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Question 1 1 of 479 selected Nonlinear Functions M

A rectangle has a length that is 15 times its width. The function y=(15w)(w) represents this situation, where y is the area, in square feet, of the rectangle and y>0. Which of the following is the best interpretation of 15w in this context?

  1. The length of the rectangle, in feet

  2. The area of the rectangle, in square feet

  3. The difference between the length and the width of the rectangle, in feet

  4.  The width of the rectangle, in feet

Show Answer Correct Answer: A

Choice A is correct. It's given that a rectangle has a length that is 15 times its width. It's also given that the function y=(15w)(w) represents this situation, where y is the area, in square feet, of the rectangle and y>0. The area of a rectangle can be calculated by multiplying the rectangle's length by its width. Since the rectangle has a length that is 15 times its width, it follows that w represents the width of the rectangle, in feet, and 15 w represents the length of the rectangle, in feet. Therefore, the best interpretation of 15 w in this context is that it's the length of the rectangle, in feet.

Choice B is incorrect. This is the best interpretation of y , not 15 w , in the given function.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect. This is the best interpretation of w , not 15 w , in the given function.

Question 2 2 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

v=-w150x

The given equation relates the distinct positive numbers v , w , and x . Which equation correctly expresses w in terms of v and x ?

  1. w = - 150 v x

  2. w=-150vx

  3. w=-x150v

  4. w = v + 150 x

Show Answer Correct Answer: A

Choice A is correct. It’s given that x is positive. Therefore, multiplying each side of the given equation by - 150 x yields -150xv=w, which is equivalent to w = - 150 v x . Thus, the equation w = - 150 v x correctly expresses w in terms of v and x .

Choice B is incorrect. This equation is equivalent to v = - w x 150 .

Choice C is incorrect. This equation is equivalent to v = - x 150 w .

Choice D is incorrect. This equation is equivalent to v = w - 150 x .

Question 3 3 of 479 selected Equivalent Expressions E

Which expression is equivalent to (m4q4z-1)(mq5z3), where m , q , and z are positive?

  1. m4q20z-3

  2. m 5 q 9 z 2

  3. m6q8z-1

  4. m20q12z-2

Show Answer Correct Answer: B

Choice B is correct. Applying the commutative property of multiplication, the expression (m4q4z-1)(mq5z3) can be rewritten as (m4m)(q4q5)(z-1z3). For positive values of x , (xa)(xb)=xa+b. Therefore, the expression (m4m)(q4q5)(z-1z3) can be rewritten as (m4+1)(q4+5)(z-1+3), or m5q9z2.

Choice A is incorrect and may result from multiplying, not adding, the exponents.

Choice C is incorrect and may result from conceptual or calculation errors. 

Choice D is incorrect and may result from conceptual or calculation errors. 

Question 4 4 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M
The figure presents the graph of a curve and a line in the x y plane, with the origin labeled O. The numbers negative 10 through 10, in increments of 5, are indicated on each axis. The line begins slightly below the x axis and to the left of the y axis, crosses the x axis at negative 10 and crosses the y axis at approximately 2. It ends above the x axis and to the right of the y axis. The curve is in the shape of a parabola that opens upward and has its vertex at approximately negative 2 point 5 on the y axis. The parabola crosses the x axis at approximately negative 5 and 5. The line and curve intersect at the point with approximate coordinates negative 6 comma 1, and at the point with approximate coordinates 7 comma 4.

A system of equations consists of a quadratic equation and a linear equation. The equations in this system are graphed in the xy-plane above. How many solutions does this system have?

  1. 0

  2. 1

  3. 2

  4. 3

Show Answer Correct Answer: C

Choice C is correct. The solutions to a system of two equations correspond to points where the graphs of the equations intersect. The given graphs intersect at 2 points; therefore, the system has 2 solutions.

Choice A is incorrect because the graphs intersect. Choice B is incorrect because the graphs intersect more than once. Choice D is incorrect. It’s not possible for the graph of a quadratic equation and the graph of a linear equation to intersect more than twice.

Question 5 5 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

6 x minus 9 y, is greater than 12

Which of the following inequalities is equivalent to the inequality above?

  1. x minus y, is greater than 2

  2. 2 x minus 3 y, is greater than 4

  3. 3 x minus 2 y, is greater than 4

  4. 3 y minus 2 x, is greater than 2

Show Answer Correct Answer: B

Choice B is correct. Both sides of the given inequality can be divided by 3 to yield 2 x minus 3 y, is greater than 4.

Choices A, C, and D are incorrect because they are not equivalent to (do not have the same solution set as) the given inequality. For example, the ordered pair 0 comma negative 1 point 5 is a solution to the given inequality, but it is not a solution to any of the inequalities in choices A, C, or D.

 

Question 6 6 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

y equals, x plus 1, 
and,
y equals, x squared plus x

If the ordered pair x comma y is a solution to the system of equations above, which of the following could be the value of x ?

  1. –1

  2. 0

  3. 2

  4. 3

Show Answer Correct Answer: A

Choice A is correct. It is given that y = x + 1 and y = x2 + x. Setting the values for y equal to each other yields x + 1 = x2 + x. Subtracting x from each side of this equation yields x2 = 1. Therefore, x can equal 1 or –1. Of these, only –1 is given as a choice.

Choice B is incorrect. If x = 0, then x + 1 = 1, but x2 + x = 02 + 0 = 0 ≠︀ 1. Choice C is incorrect. If x = 2, then x + 1 = 3, but x2 + x = 22 + 2 = 6 ≠︀ 3. Choice D is incorrect. If x = 3, then x + 1 = 4, but x2 + x = 32 + 3 = 12 ≠︀ 4.

 

Question 7 7 of 479 selected Equivalent Expressions E

Which expression is equivalent to 16 x 3 y 2 + 14 x y ?

  1. 2 x y ( 8 x y + 7 )

  2. 2 x y ( 8 x 2 y + 7 )

  3. 14 x y ( 2 x 2 y + 1 )

  4. 14 x y ( 8 x 2 y + 1 )

Show Answer Correct Answer: B

Choice B is correct. Since 2 x y is a common factor of each term in the given expression, the expression can be rewritten as 2xy(8x2y+7).

Choice A is incorrect. This expression is equivalent to 16x2y2+14xy.

Choice C is incorrect. This expression is equivalent to 28x3y2+14xy.

Choice D is incorrect. This expression is equivalent to 112x3y2+14xy.

Question 8 8 of 479 selected Equivalent Expressions M

4 a squared, plus 20 a b, plus 25 b squared

Which of the following is a factor of the polynomial above?

  1. a plus b

  2. 2 a plus 5 b

  3. 4 a plus 5 b

  4. 4 a plus 25 b

Show Answer Correct Answer: B

Choice B is correct. The first and last terms of the polynomial are both squares such that 4 a, squared, equals, open parenthesis, 2 a, close parenthesis, squared and 25 b squared, equals, open parenthesis, 5 b, close parenthesis, squared. The second term is twice the product of the square root of the first and last terms: 20 a, b, equals 2, times 2 a, times 5 b. Therefore, the polynomial is the square of a binomial such that 4 a, squared, plus 20 a, b, plus 25 b squared, equals, open parenthesis, 2 a, plus 5 b, close parenthesis, squared, and open parenthesis, 2 a, plus 5 b, close parenthesis is a factor.

Choice A is incorrect and may be the result of incorrectly factoring the polynomial. Choice C is incorrect and may be the result of dividing the second and third terms of the polynomial by their greatest common factor. Choice D is incorrect and may be the result of not factoring the coefficients.

 

Question 9 9 of 479 selected Equivalent Expressions M

If p equals, 3 x, plus 4 and v equals, x plus 5, which of the following is equivalent to p v, minus 2 p, plus v ?

  1. 3 x squared, plus, 12 x, plus 7

  2. 3 x squared, plus, 14 x, plus 17

  3. 3 x squared, plus, 19 x, plus 20

  4. 3 x squared, plus, 26 x, plus 33

Show Answer Correct Answer: B

Choice B is correct. It’s given that p equals, 3 x plus 4 and v equals, x plus 5. Substituting the values for p and v into the expression p v, minus 2 p, plus v yields open parenthesis, 3 x plus 4, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, minus, 2 times, open parenthesis, 3 x plus 4, close parenthesis, plus x, plus 5. Multiplying the terms open parenthesis, 3 x plus 4, close parenthesis, times, open parenthesis, x plus 5, close parenthesis yields 3 x squared, plus 4 x, plus 15 x, plus 20. Using the distributive property to rewrite negative 2 times, open parenthesis, 3 x plus 4, close parenthesis yields negative 6 x minus 8. Therefore, the entire expression can be represented as 3 x squared, plus 4 x, plus 15 x, plus 20, minus 6 x, minus 8, plus x, plus 5. Combining like terms yields 3 x squared, plus 14 x, plus 17.

Choice A is incorrect and may result from subtracting, instead of adding, the term x plus 5. Choice C is incorrect. This is the result of multiplying the terms open parenthesis, 3 x plus 4, close parenthesis, times, open parenthesis, x plus 5, close parenthesis. Choice D is incorrect and may result from distributing 2, instead of negative 2, to the term 3 x plus 4.

 

Question 10 10 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

x squared, minus x, minus 1, equals 0

What values satisfy the equation above?

  1. x equals 1 and x equals 2

  2. x equals negative one half and x equals three halves

  3. x equals, the fraction with numerator 1 plus the square root of 5, and denominator 2 and x equals, the fraction with numerator 1 minus the square root of 5, and denominator 2

  4. x equals, the fraction with numerator negative 1 plus the square root of 5, and denominator 2 and x equals, the fraction with numerator negative 1 minus the square root of 5, and denominator 2

Show Answer Correct Answer: C

Choice C is correct. Using the quadratic formula to solve the given expression yields x equals, the fraction with numerator negative, open parenthesis, negative 1, close parenthesis, plus or minus, the square root of open parenthesis, negative 1, close parenthesis, squared, minus, 4 times 1 times negative 1, end root, and denominator, 2 times 1, end fraction, which equals, the fraction with numerator 1, plus or minus, the square root of 5, and denominator 2. Therefore, x equals, the fraction with numerator 1 plus the square root of 5, and denominator 2 and x equals, the fraction with numerator 1 minus the square root of 5, and denominator 2 satisfy the given equation.

Choices A and B are incorrect and may result from incorrectly factoring or incorrectly applying the quadratic formula. Choice D is incorrect and may result from a sign error.

 

Question 11 11 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

3 x 2 - 15 x + 18 = 0

How many distinct real solutions are there to the given equation?

  1. Exactly one

  2. Exactly two

  3. Infinitely many

  4. Zero

Show Answer Correct Answer: B

Choice B is correct. The number of solutions to a quadratic equation of the form ax2+bx+c=0, where a, b, and c are constants, can be determined by the value of the discriminant, b2-4ac. If the value of the discriminant is positive, then the quadratic equation has exactly two distinct real solutions. If the value of the discriminant is equal to zero, then the quadratic equation has exactly one real solution. If the value of the discriminant is negative, then the quadratic equation has zero real solutions. In the given equation, 3x2-15x+18=0, a=3, b=-15, and c=18. Substituting 3 for a, -15 for b, and 18 for c in b2-4ac yields (-15)2-4(3)(18), or 9. Since the value of the discriminant is positive, the given equation has exactly two distinct real solutions.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 12 12 of 479 selected Nonlinear Functions H

The product of two positive integers is 462 . If the first integer is 5 greater than twice the second integer, what is the smaller of the two integers?

Show Answer Correct Answer: 14

The correct answer is 14 . Let x represent the first integer and y represent the second integer. If the first integer is 5 greater than twice the second integer, then x=2y+5. It's given that the product of the two integers is 462 ; therefore xy=462. Substituting 2y+5 for x in this equation yields (2y+5)(y)=462, which can be written as 2y2+5y=462. Subtracting 462 from each side of this equation yields 2y2+5y-462=0. The left-hand side of this equation can be factored by finding two values whose product is 2(-462), or - 924 , and whose sum is 5 . The two values whose product is - 924 and whose sum is 5 are 33 and - 28 . Thus, the equation 2y2+5y-462=0 can be rewritten as 2y2-28y+33y-462=0, which is equivalent to 2y(y-14)+33(y-14)=0, or (2y+33)(y-14)=0. By the zero product property, it follows that 2y+33=0 or y-14=0. Subtracting 33 from both sides of the equation 2y+33=0 yields 2y=-33. Dividing both sides of this equation by 2 yields y=-332. Since y is a positive integer, the value of y isn't -332. Adding 14 to both sides of the equation y-14=0 yields y=14. Substituting 14 for y in the equation xy=462 yields x(14)=462. Dividing both sides of this equation by 14 yields x=33. Therefore, the two integers are 14 and 33 , so the smaller of the two integers is 14 .
 

Question 13 13 of 479 selected Nonlinear Functions H

The function g is defined by g(x)=|x|a-14, where a<0. What is the product of g(15a) and g(7a)?

Show Answer Correct Answer: 609

The correct answer is 609. It’s given that the function g is defined by g(x)=|x|a-14, where a<0. Substituting 15a for x in function g yields g(15a)=|15a|a-14. This function can be rewritten as g(15a)=15|a|a-14, or g(15a)=15(|a|a)-14. Since a<0, it follows that |a|a=-1. Substituting -1 for |a|a in g(15a)=15(|a|a)-14 yields g(15a)=15(-1)-14, or g(15a)=-29. Similarly, substituting 7a for x in function g yields g(7a)=|7a|a-14. This function can be rewritten as g(7a)=7|a|a-14, or g(7a)=7(|a|a)-14. Since a<0, it again follows that |a|a=-1. Substituting -1 for |a|a in g(7a)=7(|a|a)-14 yields g(7a)=7(-1)-14, or g(7a)=-21. Therefore, g(15a)=-29 and g(7a)=-21. Thus, the product of g(15a) and g(7a) is (-29)(-21), or 609.

Question 14 14 of 479 selected Nonlinear Functions E

4812162024x50100150200250300350yONumber of monthsafter purchaseValue (dollars)
  • Moving from left to right:
    • The curve is in quadrant 1.
    • The curve trends down gradually.
  • The curve passes through the following approximate points:
    • (0 comma 225)
    • (24 comma 50)

The graph shown gives the estimated value, in dollars, of a tablet as a function of the number of months since it was purchased. What is the best interpretation of the y-intercept of the graph in this context?

  1. The estimated value of the tablet was $225 when it was purchased.

  2. The estimated value of the tablet 24 months after it was purchased was $225.

  3. The estimated value of the tablet had decreased by $225 in the 24 months after it was purchased.

  4. The estimated value of the tablet decreased by approximately 2.25% each year after it was purchased.

Show Answer Correct Answer: A

Choice A is correct. It's given that the graph shown gives the estimated value y , in dollars, of a tablet as a function of the number of months since it was purchased, x . The y-intercept of a graph is the point at which the graph intersects the y-axis, or when x is 0 . The graph shown intersects the y-axis at the point (0,225). It follows that 0 months after the tablet was purchased, or when the tablet was purchased, the estimated value of the tablet was 225 dollars. Therefore, the best interpretation of the y-intercept is that the estimated value of the tablet was $225 when it was purchased.

Choice B is incorrect. The estimated value of the tablet 24 months after it was purchased was $50, not $225.

Choice C is incorrect. The estimated value of the tablet had decreased by $225-$50, or $175, not $225, in the 24 months after it was purchased.

Choice D is incorrect and may result from conceptual errors.

Question 15 15 of 479 selected Equivalent Expressions E

Which of the following is equivalent to the given expression?

  1. 3 x minus 2

  2. 3 x plus 2

  3. 3 x minus 8

  4. 3 x plus 8

Show Answer Correct Answer: B

Choice B is correct. Using the associative and commutative properties of addition, the given expression open parenthesis, x plus 5, close parenthesis, plus, open parenthesis, 2 x minus 3, close parenthesis can be rewritten as open parenthesis, x plus 2 x, close parenthesis, plus, open parenthesis, 5 minus 3, close parenthesis. Adding these like terms results in 3 x plus 2.

Choice A is incorrect and may result from adding open parenthesis, x minus 5, close parenthesis, plus, open parenthesis, 2 x plus 3, close parenthesis. Choice C is incorrect and may result from adding open parenthesis, x minus 5, close parenthesis, plus, open parenthesis, 2 x minus 3, close parenthesis. Choice D is incorrect and may result from adding open parenthesis, x plus 5, close parenthesis, plus, open parenthesis, 2 x plus 3, close parenthesis.

 

Question 16 16 of 479 selected Nonlinear Functions H
x g(x)
-27 3
-9 0
21 5

The table shows three values of x and their corresponding values of g(x), where g(x)=f(x)x+3 and f is a linear function. What is the y-intercept of the graph of y=f(x) in the xy-plane?

  1. (0,36)

  2. (0,12)

  3. (0,4)

  4. (0,-9)

Show Answer Correct Answer: A

Choice A is correct. It's given that the table shows values of x and their corresponding values of g(x), where g(x)=f(x)x+3. It's also given that f is a linear function. It follows that an equation that defines f can be written in the form f(x)=mx+b, where m represents the slope and b represents the y-coordinate of the y-intercept (0,b) of the graph of y=f(x) in the xy-plane. The slope of the graph of y=f(x) can be found using two points, (x1,y1) and (x2,y2), that are on the graph of y=f(x), and the formula m=y2-y1x2-x1. Since the table shows values of x and their corresponding values of g(x), substituting values of x and g(x) in the equation g(x)=f(x)x+3 can be used to define function f . Using the first pair of values from the table, x=-27 and g(x)=3, yields 3=f(-27)-27+3, or 3=f(-27)-24. Multiplying each side of this equation by -24 yields -72=f(-27), so the point (-27,-72) is on the graph of y=f(x). Using the second pair of values from the table, x=-9 and g(x)=0, yields 0=f(-9)-9+3, or 0=f(-9)-6. Multiplying each side of this equation by -6 yields 0=f(-9), so the point (-9,0) is on the graph of y=f(x). Substituting (-27,-72) and (-9,0) for (x1,y1) and (x2,y2), respectively, in the formula m=y2-y1x2-x1 yields m=0-(-72)-9-(-27), or m = 4 . Substituting 4 for m in the equation f(x)=mx+b yields f(x)=4x+b. Since 0=f(-9), substituting -9 for x and 0 for f(x) in the equation f(x)=4x+b yields 0=4(-9)+b, or 0=-36+b. Adding 36 to both sides of this equation yields 36=b. It follows that 36 is the y-coordinate of the y-intercept (0,b) of the graph of y=f(x). Therefore, the y-intercept of the graph of y=f(x) is (0,36).

Choice B is incorrect. 12 is the y-coordinate of the y-intercept of the graph of y=g(x).

Choice C is incorrect. 4 is the slope of the graph of y=f(x).

Choice D is incorrect. -9 is the x-coordinate of the x-intercept of the graph of y=f(x).

Question 17 17 of 479 selected Equivalent Expressions M

Which expression is equivalent to 8x(x-7)-3(x-7)2x-14, where x>7?

  1. x-75

  2. 8x-32

  3. 8x2-3x-142x-14

  4. 8x2-3x-772x-14

Show Answer Correct Answer: B

Choice B is correct. The given expression has a common factor of 2 in the denominator, so the expression can be rewritten as 8x(x-7)-3(x-7)2(x-7). The three terms in this expression have a common factor of (x-7). Since it's given that x>7, x can't be equal to 7 , which means (x-7) can't be equal to 0 . Therefore, each term in the expression, 8x(x-7)-3(x-7)2(x-7), can be divided by (x-7), which gives 8x-32.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 18 18 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

x=49

y=x+9

The graphs of the given equations intersect at the point (x,y) in the xy-plane. What is the value of y ?

  1. 16

  2. 40

  3. 81

  4. 130

Show Answer Correct Answer: A

Choice A is correct. It's given that the graphs of the given equations intersect at the point (x,y) in the xy-plane. It follows that (x,y) represents a solution to the system consisting of the given equations. The first equation given is x = 49 . Substituting 49 for x in the second equation given, y = x + 9 , yields y=49+9, which is equivalent to y=7+9, or y=16. It follows that the graphs of the given equations intersect at the point (49,16). Therefore, the value of y is 16 .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 19 19 of 479 selected Nonlinear Functions H

h of x equals, 2 times, open parenthesis, x minus 4, close parenthesis, squared, minus 32

The quadratic function h is defined as shown. In the xy-plane, the graph of y equals, h of x intersects the x-axis at the points with coordinates 0 comma 0 and t comma 0, where t is a constant. What is the value of t ?

  1. 1

  2. 2

  3. 4

  4. 8

Show Answer Correct Answer: D

Choice D is correct. It’s given that the graph of y equals, h of x intersects the x-axis at the point with coordinates 0 comma 0 and the point with coordinates t comma 0 , where t is a constant. Since this graph intersects the x-axis when y equals 0 or when h of x equals 0, it follows that h of 0 equals 0 and h of t equals 0. If h of t equals 0, then 0 equals, 2 times, open parenthesis, t minus 4, close parenthesis, squared, minus 32. Adding 32 to both sides of this equation yields 32 equals, 2 times, open parenthesis, t minus 4, close parenthesis, squared. Dividing both sides of this equation by 2 yields 16 equals, open parenthesis, t minus 4, close parenthesis, squared. Taking the square root of both sides of this equation yields 4 equals, t minus 4. Adding 4 to both sides of this equation yields 8 equals t. Therefore, the value of t is 8.

Choices A, B, and C are incorrect and may result from calculation errors.

 

Question 20 20 of 479 selected Nonlinear Functions E

  • The parabola opens upward.
  • The vertex is at the point (2 comma negative 2).
  • The parabola passes through the following points:
    • (1 comma negative nine tenths)
    • (2 comma negative 2)
    • (3 comma negative nine tenths)

The graph shown will be translated up 4 units. Which of the following will be the resulting graph?

    • The parabola opens upward.
    • The vertex is at the point (2 comma 2).
    • The parabola passes through the following points:
      • (1 comma StartFraction 31 Over 10 EndFraction)
      • (2 comma 2)
      • (3 comma StartFraction 31 Over 10 EndFraction)

    • The parabola opens upward.
    • The vertex is at the point (2 comma negative 6).
    • The parabola passes through the following points:
      • (1 comma negative StartFraction 49 Over 10 EndFraction)
      • (2 comma negative 6)
      • (3 comma negative StartFraction 49 Over 10 EndFraction)

    • The parabola opens upward.
    • The vertex is at the point (negative 2 comma negative 2).
    • The parabola passes through the following points:
      • (negative 3 comma negative nine tenths)
      • (negative 2 comma negative 2)
      • (negative 1 comma negative nine tenths)

    • The parabola opens upward.
    • The vertex is at the point (6 comma negative 2).
    • The parabola passes through the following points:
      • (5 comma negative nine tenths)
      • (6 comma negative 2)
      • (7 comma negative nine tenths)

Show Answer Correct Answer: A

Choice A is correct. When a graph is translated up 4 units, each point on the resulting graph is 4 units above the point on the original graph. In other words, the y-value of each point on the graph increases by 4 . The graph shown passes through the points (1,-1), (2,-2), and (3,-1). It follows that when the graph shown is translated up 4 units, the resulting graph will pass through the points (1,-1+4), (2,-2+4), and (3,-1+4). These points are (1,3)(2,2), and (3,3), respectively. Of the given choices, only the graph in choice A passes through the points (1,3)(2,2), and (3,3).

Choice B is incorrect. This is the result of translating the graph down, rather than up, 4 units.

Choice C is incorrect. This is the result of translating the graph left, rather than up, 4 units.

Choice D is incorrect. This is the result of translating the graph right, rather than up, 4 units.

Question 21 21 of 479 selected Nonlinear Functions H

f(x)= x 2 - 48 x + 2,304

What is the minimum value of the given function?

Show Answer Correct Answer: 1728

The correct answer is 1,728 . The given function can be rewritten in the form f(x)=a(x-h)2+k, where a is a positive constant and the minimum value, k , of the function occurs when the value of x is h . By completing the square, f(x)=x2-48x+2,304 can be written as f(x)=x2-48x+(482)2+2,304-(482)2, or f(x)=(x-24)2+1,728. This equation is in the form f(x)=a(x-h)2+k, where a = 1 , h = 24 , and k = 1,728 . Therefore, the minimum value of the given function is 1,728 .

Question 22 22 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

2|4-x|+3|4-x|=25

What is the positive solution to the given equation?

Show Answer Correct Answer: 9

The correct answer is 9 . The given equation can be rewritten as 5|4-x|=25. Dividing each side of this equation by 5 yields |4-x|=5. By the definition of absolute value, if |4-x|=5, then 4-x=5 or 4-x=-5. Subtracting 4 from each side of the equation 4-x=5 yields -x=1. Dividing each side of this equation by -1 yields x=-1. Similarly, subtracting 4 from each side of the equation 4-x=-5 yields -x=-9. Dividing each side of this equation by -1 yields x=9. Therefore, since the two solutions to the given equation are -1 and 9 , the positive solution to the given equation is 9 .

Question 23 23 of 479 selected Nonlinear Functions H

The function f is defined by f(x)=(-8)(2)x+22. What is the y-intercept of the graph of y=f(x) in the xy-plane?

  1. (0,14)

  2. (0,2)

  3. (0,22)

  4. (0,-8)

Show Answer Correct Answer: A

Choice A is correct. The y-intercept of the graph of y=f(x) in the xy-plane occurs at the point on the graph where x = 0 . In other words, when x = 0 , the corresponding value of f(x) is the y-coordinate of the y-intercept. Substituting 0 for x in the given equation yields f(0)=(-8)(2)0+22, which is equivalent to f(0)=(-8)(1)+22, or f(0)=14. Thus, when x = 0 , the corresponding value of f(x) is 14 . Therefore, the y-intercept of the graph of y=f(x) in the xy-plane is (0,14).

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This could be the y-intercept for f(x)=(-8)(2)x, not f(x)=(-8)(2)x+22.

Question 24 24 of 479 selected Nonlinear Functions E

  • All values are approximate.
  • The parabola opens downward.
  • The vertex of the parabola is at (0.6 comma 5.5).
  • The parabola passes through the following points:
    • (0.0 comma 3.8)
    • (0.6 comma 5.5)
    • (1.0 comma 4.8)
    • (1.7 comma 0.0)

The graph shows the height above ground, in meters, of a ball x seconds after the ball was launched upward from a platform. Which statement is the best interpretation of the marked point (1.0,4.8) in this context?

  1. 1.0 second after being launched, the ball's height above ground is 4.8 meters.

  2. 4.8 seconds after being launched, the ball's height above ground is 1.0 meter.

  3. The ball was launched from an initial height of 1.0 meter with an initial velocity of 4.8 meters per second.

  4. The ball was launched from an initial height of 4.8 meters with an initial velocity of 1.0 meter per second.

Show Answer Correct Answer: A

Choice A is correct. It's given that the graph shows the height above ground, in meters, of a ball x seconds after the ball was launched upward from a platform. In the graph shown, the x-axis represents time, in seconds, and the y-axis represents the height of the ball above ground, in meters. It follows that for the marked point (1.0,4.8), 1.00 represents the time, in seconds, after the ball was launched upward from a platform and 4.80 represents the height of the ball above ground, in meters. Therefore, the best interpretation of the marked point (1.0,4.8) is 1.00 second after being launched, the ball's height above ground is 4.80 meters.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 25 25 of 479 selected Nonlinear Functions H

An auditorium has seats for 1,800 people. Tickets to attend a show at the auditorium currently cost $4.00 . For each $1.00 increase to the ticket price, 100 fewer tickets will be sold. This situation can be modeled by the equation y = - 100 x 2 + 1,400 x + 7,200 , where x represents the increase in ticket price, in dollars, and y represents the revenue, in dollars, from ticket sales. If this equation is graphed in the xy-plane, at what value of x is the maximum of the graph?

  1. 4

  2. 7

  3. 14

  4. 18

Show Answer Correct Answer: B

Choice B is correct. It’s given that the situation can be modeled by the equation y=-100x2+1,400x+7,200, where x represents the increase in ticket price, in dollars, and y represents the revenue, in dollars, from ticket sales. Since the coefficient of the x2 term is negative, the graph of this equation in the xy-plane opens downward and reaches its maximum value at its vertex. If a quadratic equation in the form y=ax2+bx+c, where a, b, and c are constants, is graphed in the xy-plane, the x-coordinate of the vertex is equal to -b2a. For the equation y=-100x2+1,400x+7,200, a=-100, b=1,400, and c=7,200. It follows that the x-coordinate of the vertex is -1,4002(-100), or 7. Therefore, if the given equation is graphed in the xy-plane, the maximum of the graph occurs at an x-value of 7.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 26 26 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

x 2 - 2 x - 9 = 0

One solution to the given equation can be written as 1+k, where k is a constant. What is the value of k ?

  1. 8

  2. 10

  3. 20

  4. 40

Show Answer Correct Answer: B

Choice B is correct. Adding 9 to each side of the given equation yields x 2 - 2 x = 9 . To complete the square, adding 1 to each side of this equation yields x2-2x+1=9+1, or (x-1)2=10. Taking the square root of each side of this equation yields x-1=±10. Adding 1 to each side of this equation yields x=1±10. Since it's given that one of the solutions to the equation can be written as 1+k, the value of k must be 10 .

Alternate approach: It's given that 1+k is a solution to the given equation. It follows that x=1+k. Substituting 1+k for x in the given equation yields (1+k)2-2(1+k)-9=0, or (1+k)(1+k)-2(1+k)-9=0. Expanding the products on the left-hand side of this equation yields 1+2k+k-2-2k-9=0, or k - 10 = 0 . Adding 10 to each side of this equation yields k = 10 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 27 27 of 479 selected Nonlinear Functions M

p(x)+57=x2

The given equation relates the value of x and its corresponding value of p(x) for the function p . What is the minimum value of the function p ?

  1. -3,249

  2. -57

  3. 57

  4. 3,249

Show Answer Correct Answer: B

Choice B is correct. For a quadratic function defined by an equation of the form p(x)=a(x-h)2+k, where a , h , and k are constants and a>0, the minimum value of the function is k . Subtracting 57 from both sides of the given equation yields p(x)=x2-57. This function is in the form p(x)=a(x-h)2+k, where a = 1 , h = 0 , and k = - 57 . Therefore, the minimum value of the function p is - 57 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 28 28 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

In the x y -plane, a line with equation 2 y = 4.5 intersects a parabola at exactly one point. If the parabola has equation y = - 4 x 2 + b x , where b is a positive constant, what is the value of b ?

Show Answer Correct Answer: 6

The correct answer is 6 . It’s given that a line with equation 2y=4.5 intersects a parabola with equation y=-4x2+bx, where b is a positive constant, at exactly one point in the xy-plane. It follows that the system of equations consisting of 2y=4.5 and y=-4x2+bx has exactly one solution. Dividing both sides of the equation of the line by 2 yields y=2.25. Substituting 2.25 for y in the equation of the parabola yields 2.25=-4x2+bx. Adding 4x2 and subtracting bx from both sides of this equation yields 4x2-bx+2.25=0. A quadratic equation in the form of ax2+bx+c=0, where a , b , and c are constants, has exactly one solution when the discriminant, b2-4ac, is equal to zero. Substituting 4 for a and 2.25 for c in the expression b2-4ac and setting this expression equal to 0 yields b2-4(4)(2.25)=0, or b2-36=0. Adding 36 to each side of this equation yields b2=36. Taking the square root of each side of this equation yields b=±6. It’s given that b is positive, so the value of b is 6 .

Question 29 29 of 479 selected Nonlinear Functions H

f(x)=9,000(0.66)x

The given function f models the number of advertisements a company sent to its clients each year, where x represents the number of years since 1997, and 0x5. If y=f(x) is graphed in the xy-plane, which of the following is the best interpretation of the y-intercept of the graph in this context?

  1. The minimum estimated number of advertisements the company sent to its clients during the 5 years was 1,708 .

  2. The minimum estimated number of advertisements the company sent to its clients during the 5 years was 9,000 .

  3. The estimated number of advertisements the company sent to its clients in 1997 was 1,708 .

  4. The estimated number of advertisements the company sent to its clients in 1997 was 9,000 .

Show Answer Correct Answer: D

Choice D is correct. The y-intercept of a graph in the xy-plane is the point where x = 0 . For the given function f, the y-intercept of the graph of y=f(x) in the xy-plane can be found by substituting 0 for x in the equation y=9,000(0.66)x, which gives y=9,000(0.66)0. This is equivalent to y=9,000(1), or y = 9,000 . Therefore, the y-intercept of the graph of y=f(x) is (0,9,000). It’s given that the function f models the number of advertisements a company sent to its clients each year. Therefore, f(x) represents the estimated number of advertisements the company sent to its clients each year. It's also given that x represents the number of years since 1997. Therefore, x = 0 represents 0 years since 1997, or 1997. Thus, the best interpretation of the y-intercept of the graph of y=f(x) is that the estimated number of advertisements the company sent to its clients in 1997 was 9,000 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 30 30 of 479 selected Equivalent Expressions E

Which expression is equivalent to (9x3+5x+7)+(6x3+5x2-5)?

  1. 15 x 6 + 5 x 2 - 5 x - 35

  2. 15 x 3 + 10 x 2 + 2

  3. 15 x 6 + 5 x 2 + 5 x + 2

  4. 15 x 3 + 5 x 2 + 5 x + 2

Show Answer Correct Answer: D

Choice D is correct. The given expression can be rewritten as (9x3+6x3)+5x2+5x+(7-5). Combining like terms in this expression yields 15x3+5x2+5x+2.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 31 31 of 479 selected Nonlinear Functions H

The first term of a sequence is 9 . Each term after the first is 4 times the preceding term. If w represents the n th term of the sequence, which equation gives w in terms of n ?

  1. w=4(9n)

  2. w=4(9n-1)

  3. w=9(4n)

  4. w=9(4n-1)

Show Answer Correct Answer: D

Choice D is correct. Since w represents the n th term of the sequence and 9 is the first term of the sequence, the value of w is 9 when the value of n is 1 . Since each term after the first is 4 times the preceding term, the value of w is 9(4) when the value of n is 2 . Therefore, the value of w is 9(4)(4), or 9(4)2, when the value of n is 3 . More generally, the value of w is 9(4n-1) for a given value of n . Therefore, the equation w=9(4n-1) gives w in terms of n .

Choice A is incorrect. This equation describes a sequence for which the first term is 36 , rather than 9 , and each term after the first is 9 , rather than 4 , times the preceding term. 

Choice B is incorrect. This equation describes a sequence for which the first term is 4 , rather than 9 , and each term after the first is 9 , rather than 4 , times the preceding term.

Choice C is incorrect. This equation describes a sequence for which the first term is 36 , rather than 9 .

Question 32 32 of 479 selected Nonlinear Functions H

The function f is defined by f(x)=a(2.2x+2.2b), where a and b are integer constants and 0<a<b. The functions g and h are equivalent to function f , where k and m are constants. Which of the following equations displays the y-coordinate of the y-intercept of the graph of y=f(x) in the xy-plane as a constant or coefficient?

  1. g(x)=a(2.2x+k)
  2. h(x)=a(2.2)x+m 
  1. I only

  2. II only

  3. I and II

  4. Neither I nor II

Show Answer Correct Answer: D

Choice D is correct. A y-intercept of a graph in the xy-plane is a point where the graph intersects the y-axis, or a point where x = 0 . Substituting 0 for x in the equation defining function f yields f(0)=a(2.20+2.2b), or f(0)=a(1+2.2b). So, the y-coordinate of the y-intercept of the graph is a(1+2.2b), or equivalently, a+a(2.2)b. It's given that function g is equivalent to function f , where 0<a<b. It follows that k=2.2b. Since a(2.2)b can't be equal to 0 , the coefficient a can't be equal to a+a(2.2)b. Since 0<a, the constant k , which is equal to 2.2b, can't be equal to a+a(2.2)b. Therefore, function g doesn't display the y-coordinate of the y-intercept of the graph of y=f(x) in the xy-plane as a constant or coefficient. It's also given that function h is equivalent to function f , where 0<a<b. The equation defining f can be rewritten as f(x)=a(2.2)x+a(2.2)b. It follows that m=a(2.2)b. Since a(2.2)b can't be equal to 0 , the coefficient a can't be equal to a+a(2.2)b. Since 0<a, the constant m , which is equal to a(2.2)b, can't be equal to a+a(2.2)b. Therefore, function h doesn't display the y-coordinate of the y-intercept of the graph of y=f(x) in the xy-plane as a constant or coefficient. Thus, neither function g nor function h displays the y-coordinate of the y-intercept of the graph of y=f(x) in the xy-plane as a constant or coefficient.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 33 33 of 479 selected Nonlinear Functions M

A physics class is planning an experiment about a toy rocket. The equation y=-16(x-5.6)2+502 gives the estimated height y , in feet, of the toy rocket x seconds after it is launched into the air. Which of the following is the best interpretation of the vertex of the graph of the equation in the xy-plane?

  1. This toy rocket reaches an estimated maximum height of 502 feet 16 seconds after it is launched into the air.

  2. This toy rocket reaches an estimated maximum height of 502 feet 5.6 seconds after it is launched into the air.

  3. This toy rocket reaches an estimated maximum height of 16 feet 502 seconds after it is launched into the air.

  4. This toy rocket reaches an estimated maximum height of 5.6 feet 502 seconds after it is launched into the air.

Show Answer Correct Answer: B

Choice B is correct. The vertex of the graph of a quadratic equation is where it reaches its minimum or maximum value. When a quadratic equation is written in the form y=a(x-h)2+k, the vertex of the parabola represented by the equation is (x,y)=(h,k). In the given equation y=-16(x-5.6)2+502, the value of h is 5.6 and the value of k is 502 . It follows that the vertex of the graph of this equation in the xy-plane is (x,y)=(5.6,502). Additionally, since a = - 16 in the given equation, the graph of the quadratic equation opens down, and the vertex represents a maximum. It’s given that the value of y represents the estimated height, in feet, of the toy rocket x seconds after it is launched into the air. Therefore, this toy rocket reaches an estimated maximum height of 502 feet 5.6 seconds after it is launched into the air.

Choice A is incorrect. The 16 in the equation is an indicator of how narrow the graph of the equation is rather than where it reaches its maximum.

Choice C is incorrect. The 16 in the equation is an indicator of how narrow the graph of the equation is rather than where it reaches its maximum.

Choice D is incorrect. This is an interpretation of the vertex of the graph of the equation y=-16(x-502)2+5.6, not of the equation y=-16(x-5.6)2+502.

Question 34 34 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

Equation 1: x minus y, equals 1. Equation 2: x plus y equals, x squared minus 3

Which ordered pair is a solution to the system of equations above?

  1. 1 plus the square root of 3, comma, the square root of 3

  2. the square root of 3, comma, the negative of the square root of 3

  3. 1 plus the square root of 5, comma, the square root of 5

  4. the square root of 5, comma, negative 1 plus the square root of 5

Show Answer Correct Answer: A

Choice A is correct. The solution to the given system of equations can be found by solving the first equation for x, which gives x equals, y plus 1, and substituting that value of x into the second equation which gives y plus 1, plus y, equals, open parenthesis, y plus 1, close parenthesis, squared, minus 3. Rewriting this equation by adding like terms and expanding open parenthesis, y plus 1, close parenthesis, squared gives 2 y plus 1, equals, y squared plus 2 y, minus 2. Subtracting 2 y from both sides of this equation gives 1 equals, y squared minus 2. Adding to 2 to both sides of this equation gives 3 equals y squared. Therefore, it follows that y equals, plus or minus the square root of 3. Substituting the square root of 3 for y in the first equation yields x minus the square root of 3, equals 1. Adding the square root of 3 to both sides of this equation yields x equals, 1 plus the square root of 3. Therefore, the ordered pair 1 plus the square root of 3, end root, comma the square root of 3 is a solution to the given system of equations.

Choice B is incorrect. Substituting the square root of 3 for x and the negative of the square root of 3 for y in the first equation yields the square root of 3, end root, minus, the negative of the square root of 3, equals 1, or 2 times the square root of 3, equals 1, which isn’t a true statement. Choice C is incorrect. Substituting 1 plus the square root of 5 for x and the square root of 5 for y in the second equation yields open parenthesis, 1 plus the square root of 5, close parenthesis, plus the square root of 5, equals, open parenthesis, 1 plus the square root of 5, close parenthesis, squared, minus 3, or 1 plus, 2 times the square root of 5, equals, 2 times the square root of 5, plus 3, which isn’t a true statement. Choice D is incorrect. Substituting the square root of 5 for x and open parenthesis, negative 1 plus the square root of 5, close parenthesis for y in the second equation yields the square root of 5 plus, open parenthesis, negative 1 plus the square root of 5, close parenthesis, equals, the square root of 5, squared, minus 3, or 2 times the square root of 5, minus 1, equals 2, which isn’t a true statement.

 

Question 35 35 of 479 selected Nonlinear Functions E

  • Moving from left to right:
    • The curve passes from quadrant 3 to quadrant 4.
    • In quadrant 3, the curve trends up sharply to point (0 comma negative 5).
    • In quadrant 4, the curve trends up gradually.
  • As x increases, the curve approaches the line y equals negative 4.
  • The curve passes through the following points:
    • (negative 1 comma negative 9)
    • (0 comma negative 5)
    • (1 comma negative 4.2)

What is the y -intercept of the graph shown?

  1. (-1,-9)

  2. (0,-5)

  3. (0,-4)

  4. (0,0)

Show Answer Correct Answer: B

Choice B is correct. The y-intercept of a graph in the xy-plane is the point (x,y) on the graph where x = 0 . At x = 0 , the corresponding value of y is -5 . Therefore, the y-intercept of the graph shown is (0,-5).

Choice A is incorrect and may result from conceptual errors.

Choice C is incorrect. This is the y-intercept of a graph in the xy-plane that intersects the y-axis at y = -4 , not y = -5 .

Choice D is incorrect. This is the y-intercept of a graph in the xy-plane that intersects the y-axis at y = 0 , not y = -5 .

Question 36 36 of 479 selected Equivalent Expressions M

Which of the following is equivalent to the expression x to the fourth power, minus x squared, minus 6 ?

  1. open parenthesis, x squared plus 1, close parenthesis, times, open parenthesis, x squared minus 6, close parenthesis

  2. open parenthesis, x squared plus 2, close parenthesis, times, open parenthesis, x squared minus 3, close parenthesis

  3. open parenthesis, x squared plus 3, close parenthesis, times, open parenthesis, x squared minus 2, close parenthesis

  4. open parenthesis, x squared plus 6, close parenthesis, times, open parenthesis, x squared minus 1, close parenthesis

Show Answer Correct Answer: B

Choice B is correct. The term x4 can be factored as x squared, times, x squared. Factoring –6 as 2 times negative 3 yields values that add to –1, the coefficient of x2 in the expression.

Choices A, C, and D are incorrect and may result from finding factors of –6 that don’t add to the coefficient of x2 in the original expression.

 

Question 37 37 of 479 selected Equivalent Expressions M

open parenthesis, 2 x plus 5, close parenthesis, squared, minus, open parenthesis, x minus 2, close parenthesis, plus 2 times, open parenthesis, x plus 3, close parenthesis

Which of the following is equivalent to the expression above?

  1. 4 x squared, plus 21 x, plus 33

  2. 4 x squared, plus 21 x, plus 29

  3. 4 x squared, plus x, plus 29

  4. 4 x squared, plus x, plus 33

Show Answer Correct Answer: A

Choice A is correct. The given expression can be rewritten as open parenthesis, 2 x plus 5, close parenthesis, squared, plus, negative 1 times, open parenthesis, x minus 2, close parenthesis, plus, 2 times, open parenthesis, x plus 3, close parenthesis. Applying the distributive property, the expression  negative 1 times, open parenthesis, x minus 2, close parenthesis, plus, 2 times, open parenthesis, x plus 3, close parenthesiscan be rewritten as negative one times x, plus, negative 1 times negative 2, plus, 2 times x, plus, 2 times 3, or negative x plus 2, plus 2 x, plus 6. Adding like terms yields x plus 8. Substituting x plus 8 for  negative 1, times, open parenthesis, x minus 2, close parenthesis, plus, 2 times, open parenthesis, x plus 3, close parenthesis in the given expression yields open parenthesis, 2 x plus 5, close parenthesis, squared, plus x, plus 8. By the rules of exponents, the expression open parenthesis, 2 x plus 5, close parenthesis, squared is equivalent to open parenthesis, 2 x plus 5, close parenthesis, times, open parenthesis, 2 x plus 5, close parenthesis. Applying the distributive property, this expression can be rewritten as 2 x times 2x, plus, 2 x times 5, plus, 5 times 2 x, plus, 5 times 5, or 4 x squared, plus 10 x, plus 10 x, plus 25. Adding like terms gives 4 x squared, plus 20 x, plus 25. Substituting 4 x squared, plus 20 x, plus 25 for open parenthesis, 2 x plus 5, close parenthesis, squared in the rewritten expression yields 4 x squared, plus 20 x, plus 25, plus x, plus 8, and adding like terms yields 4 x squared, plus 21 x, plus 33.

Choices B, C, and D are incorrect. Choices C and D may result from rewriting the expression open parenthesis, 2 x plus 5, close parenthesis, squared as 4 x squared, plus 25, instead of as 4 x squared, plus 20 x, plus 25. Choices B and C may result from rewriting the expression negative, open parenthesis, x minus 2, close parenthesis as negative x minus 2, instead of negative x plus 2.

 

Question 38 38 of 479 selected Nonlinear Functions M
Time (years) Total amount (dollars)
0 604.00
1 606.42
2 608.84

Rosa opened a savings account at a bank. The table shows the exponential relationship between the time t , in years, since Rosa opened the account and the total amount n , in dollars, in the account. If Rosa made no additional deposits or withdrawals, which of the following equations best represents the relationship between t and n ?

  1. n=(1+604)t

  2. n=(1+0.004)t

  3. n=604(1+0.004)t

  4. n=0.004(1+604)t

Show Answer Correct Answer: C

Choice C is correct. It’s given that the relationship between t and n is exponential. The table shows that the value of n increases as the value of t increases. Therefore, the relationship between t and n can be represented by an increasing exponential equation of the form n=a(1+b)t, where a and b are positive constants. The table shows that when t = 0 , n = 604 . Substituting 0 for t and 604 for n in the equation n=a(1+b)t yields 604=a(1+b)0, which is equivalent to 604=a(1), or 604=a. Substituting 604 for a in the equation n=a(1+b)t yields n=604(1+b)t. The table also shows that when t = 1 , n = 606.42 . Substituting 1 for t and 606.42 for n in the equation n=604(1+b)t yields 606.42=604(1+b)1, or 606.42=604(1+b). Dividing both sides of this equation by 604 yields approximately 1.004=1+b. Subtracting 1 from both sides of this equation yields that the value of b is approximately 0.004 . Substituting 0.004 for b in the equation n=604(1+b)t yields n=604(1+0.004)t. Therefore, of the choices, choice C best represents the relationship between t and n .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 39 39 of 479 selected Equivalent Expressions H

open parenthesis, a x plus 3, close parenthesis, times, open parenthesis, 5 x squared, minus b x, plus 4, close parenthesis, equals, 20 x cubed, minus 9 x squared, minus 2 x, plus 12

The equation above is true for all x, where a and b are constants. What is the value of ab ?

  1. 18

  2. 20

  3. 24

  4. 40

Show Answer Correct Answer: C

Choice C is correct. If the equation is true for all x, then the expressions on both sides of the equation will be equivalent. Multiplying the polynomials on the left-hand side of the equation gives 5 a, x cubed, minus a, b x squared, plus 4 a, x, plus 15 x squared, minus 3 b x, plus 12. On the right-hand side of the equation, the only x squared-term is negative 9 x squared. Since the expressions on both sides of the equation are equivalent, it follows that negative a, b x squared, plus 15 x squared, equals negative 9 x squared, which can be rewritten as open parenthesis, negative a, b plus 15, close parenthesis, times x squared, equals negative 9 x squared. Therefore, negative a, b plus 15, equals negative 9, which gives a, b equals 24.

Choice A is incorrect. If a, b equals 18, then the coefficient of x squared on the left-hand side of the equation would be negative 18 plus 15, equals negative 3, which doesn’t equal the coefficient of x squared, negative 9, on the right-hand side. Choice B is incorrect. If a, b equals 20, then the coefficient of x squared on the left-hand side of the equation would be negative 20 plus 15, equals negative 5, which doesn’t equal the coefficient of x squared, negative 9, on the right-hand side. Choice D is incorrect. If a, b equals 40, then the coefficient of x squared on the left-hand side of the equation would be negative 40 plus 15, equals negative 25, which doesn’t equal the coefficient of x squared, negative 9, on the right-hand side.

 

Question 40 40 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

x2-12x+27=0

How many distinct real solutions does the given equation have?

  1. Exactly two

  2. Exactly one

  3. Zero

  4. Infinitely many

Show Answer Correct Answer: A

Choice A is correct. The number of solutions of a quadratic equation of the form a x 2 + b x + c = 0 , where a , b , and c are constants, can be determined by the value of the discriminant, b2-4ac. If the value of the discriminant is positive, then the quadratic equation has exactly two distinct real solutions. If the value of the discriminant is equal to zero, then the quadratic equation has exactly one real solution. If the value of the discriminant is negative, then the quadratic equation has zero real solutions. In the given equation, x2-12x+27=0, a = 1 , b = - 12 , and c = 27 . Substituting these values for a , b , and c in b2-4ac yields (-12)2-4(1)(27), or 36 . Since the value of its discriminant is positive, the given equation has exactly two distinct real solutions.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 41 41 of 479 selected Nonlinear Functions M
x g(x)
-1 25
0 1
1 1 25
2 1 625

For the exponential function g , the table shows four values of x and their corresponding values of g(x). Which equation defines g ?

  1. g(x)=-25x

  2. g(x)=-(125)x

  3. g(x)=25x

  4. g(x)=(125)x

Show Answer Correct Answer: D

Choice D is correct. It's given that function g is exponential. Therefore, an equation defining g can be written in the form g(x)=a(b)x, where a and b are constants. The table shows that when x = 0 , g(x)=1. Substituting 0 for x and 1 for g(x) in the equation g(x)=a(b)x yields 1=a(b)0, which is equivalent to 1 = a . Substituting 1 for a in the equation g(x)=a(b)x yields g(x)=(b)x. The table also shows that when x = 1 , g(x)=125. Substituting 1 for x and 1 25 for g(x) in the equation g(x)=(b)x yields 125=(b)1, which is equivalent to 1 25 = b . Substituting 1 25 for b in the equation g(x)=(b)x yields g(x)=(125)x.

Choice A is incorrect. For this function, g(1) is equal to - 25 , not 1 25 .

Choice B is incorrect. For this function, g(1) is equal to - 1 25 , not 1 25 .

Choice C is incorrect. For this function, g(1) is equal to 25 , not 1 25 .

Question 42 42 of 479 selected Equivalent Expressions M

Which of the following expressions is equivalent to x squared minus 5 ?

  1. open parenthesis, x plus the square root of 5, close parenthesis, squared
  2. open parenthesis, x minus the square root of 5, close parenthesis, squared
  3. open parenthesis, x plus the square root of 5, close parenthesis, times, open parenthesis, x minus the square root of 5, close parenthesis
  4. open parenthesis, x plus 5, close parenthesis, times, open parenthesis, x minus 1, close parenthesis
Show Answer Correct Answer: C

Choice C is correct. The expression can be written as a difference of squares x2 – y2, which can be factored as (x + y)(x – y). Here, y2 = 5, so y equals the square root of 5, and the expression therefore factors as open parenthesis, x plus the square root of 5, close parenthesis, times, open parenthesis, x minus the square root of 5, close parenthesis.

Choices A and B are incorrect and may result from misunderstanding how to factor a difference of squares. Choice D is incorrect; (x + 5)(x – 1) can be rewritten as x2 + 4x – 5, which is not equivalent to the original expression.

Question 43 43 of 479 selected Equivalent Expressions H

Which of the following expressions is(are) a factor of 3 x 2 + 20 x - 63 ?

  1.   x - 9
  2. 3 x - 7
  1. I only

  2. II only

  3. I and II

  4. Neither I nor II

Show Answer Correct Answer: B

Choice B is correct. The given expression can be factored by first finding two values whose sum is 20 and whose product is 3(-63), or -189 . Those two values are 27 and -7 . It follows that the given expression can be rewritten as 3x2+27x-7x-63. Since the first two terms of this expression have a common factor of 3 x and the last two terms of this expression have a common factor of -7 , this expression can be rewritten as 3x(x+9)-7(x+9). Since the two terms of this expression have a common factor of (x+9), it can be rewritten as (3x-7)(x+9). Therefore, expression II, 3x-7, is a factor of 3x2+20x-63, but expression I, x-9, is not a factor of 3x2+20x-63

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 44 44 of 479 selected Equivalent Expressions H

If the fraction with numerator the square root of x to the fifth power, end root, and denominator the cube root of x to the fourth power, end root, end fraction, equals x raised to the fraction a over b power for all positive values of x, what is the value of the fraction a over b?

Show Answer

The correct answer is 7 over 6. The value of a, over b can be found by first rewriting the left-hand side of the given equation as the fraction with numerator x raised to the five halves power, and denominator x raised to the four thirds power, end fraction . Using the properties of exponents, this expression can be rewritten as x raised to the open parenthesis, five halves, minus four thirds, close parenthesis, power. This expression can be rewritten by subtracting the fractions in the exponent, which yields x raised to the fraction 7 over 6, power. Thus, a, over b is 7 over 6. Note that 7/6, 1.166, and 1.167 are examples of ways to enter a correct answer.

Question 45 45 of 479 selected Nonlinear Functions M
x h(x)
0 1.23
2 1.54
4 1.94

The table shows the exponential relationship between the number of years, x , since Hana started training in pole vault, and the estimated height h(x), in meters, of her best pole vault for that year. Which of the following functions best represents this relationship, where x4?

  1. h(x)=1.12(0.23)x

  2. h(x)=1.12(1.23)x

  3. h(x)=1.23(0.12)x

  4. h(x)=1.23(1.12)x

Show Answer Correct Answer: D

Choice D is correct. The table shows an increasing exponential relationship between the number of years, x , since Hana started training in pole vault and the estimated height h(x), in meters, of her best pole vault for that year. The relationship can be written as h(x)=Cax, where C and a are positive constants. It's given that when x = 0 , h(x)=1.23. Substituting 0 for x and 1.23 for h(x) in h(x)=Cax yields 1.23=Ca0, or 1.23=C. Substituting 1.23 for C in h(x)=Cax yields h(x)=1.23ax. It's also given that when x = 2 , h(x)=1.54. Substituting 2 for x and 1.54 for h(x) in h(x)=1.23ax yields 1.54=1.23a2. Dividing each side of this equation by 1.23 yields 1.541.23=1.23a21.23, or a 2 is approximately equal to 1.252 . Since a is positive, a is approximately equal to 1.252, or 1.12 . Substituting 1.12 for a in h(x)=1.23ax yields h(x)=1.23(1.12)x.

Choice A is incorrect. When x = 0 , the value of h(x) in this function is equal to 1.12 rather than 1.23 , and it is decreasing rather than increasing.

Choice B is incorrect. When x = 0 , the value of h(x) in this function is equal to 1.12 rather than 1.23 .

Choice C is incorrect. This function is decreasing rather than increasing.

Question 46 46 of 479 selected Nonlinear Functions H

Function f is defined by f(x)=-ax+b, where a and b are constants. In the xy-plane, the graph of y=f(x)-12 has a y-intercept at (0,-757). The product of a and b is 320 7 . What is the value of a ?

Show Answer Correct Answer: 20

The correct answer is 20 . It’s given that f(x)=-ax+b. Substituting -ax+b for f(x) in the equation y=f(x)-12 yields y=-ax+b-12. It’s given that the y-intercept of the graph of y=f(x)-12 is (0,-757). Substituting 0 for x and -757 for y in the equation y=-ax+b-12 yields -757=-a0+b-12, which is equivalent to -757=-1+b-12, or -757=b-13. Adding 13 to both sides of this equation yields 167=b. It’s given that the product of a and b is 320 7 , or a b = 320 7 . Substituting 16 7 for b in this equation yields (a)(167)=3207. Dividing both sides of this equation by 16 7 yields a = 20 .

Question 47 47 of 479 selected Nonlinear Functions H

f(x)=(x-44)(x-46)

The function f is defined by the given equation. For what value of x does f(x) reach its minimum? 

  1. 46

  2. 45

  3. 44

  4. -1

Show Answer Correct Answer: B

Choice B is correct. It's given that f(x)=(x-44)(x-46), which can be rewritten as f(x)=x2-90x+2,024. Since the coefficient of the x2-term is positive, the graph of y=f(x) in the xy-plane opens upward and reaches its minimum value at its vertex. For an equation in the form f(x)=ax2+bx+c, where a , b , and c are constants, the x-coordinate of the vertex is -b2a. For the equation f(x)=x2-90x+2,024, a = 1 , b = - 90 , and c = 2,024 . It follows that the x-coordinate of the vertex is -(-90)2(1), or 45 . Therefore, f(x) reaches its minimum when the value of x is 45 .

Choice A is incorrect. This is one of the x-coordinates of the x-intercepts of the graph of y=f(x) in the xy-plane.

Choice C is incorrect. This is one of the x-coordinates of the x-intercepts of the graph of y=f(x) in the xy-plane.

Choice D is incorrect. This is the y-coordinate of the vertex of the graph of y=f(x) in the xy-plane.

Question 48 48 of 479 selected Nonlinear Functions M

f(x)=4,000(0.75)x

An entomologist recommended a program to reduce a certain invasive beetle population in an area. The given function estimates this beetle species' population x years after 2012, where x7. Which of the following is the best interpretation of 4,000 in this context?

  1. The estimated initial beetle population for this species and area in 2012

  2. The estimated beetle population for this species and area 7 years after 2012

  3. The estimated percent decrease in the beetle population for this species and area each year after 2012

  4. The estimated percent decrease in the beetle population for this species and area every 7 years after 2012

Show Answer Correct Answer: A

Choice A is correct. For an exponential function in the form f(x)=a(b)x, where a and b are positive constants and b<1, the initial value of f(x), or the value of f(x) when x=0, is a and the value of f(x) decreases by 100(1-b)% each time x increases by 1. Therefore, the initial value of the function f(x)=4,000(0.75)x, or the value of f(x) when x=0, is 4,000. Therefore, the best interpretation of 4,000 in this context is the estimated initial beetle population for this species and area in 2012.

Choice B is incorrect. The estimated beetle population for this species and area 7 years after 2012 is 4,000(0.75)7, or approximately 534, not 4,000.

Choice C is incorrect. The estimated percent decrease in the beetle population for this species and area each year after 2012 is 100(1-0.75), or 25, not 4,000.

Choice D is incorrect. The estimated percent decrease in the beetle population for this species and area every 7 years after 2012 is 100(1-0.757), or approximately 87, not 4,000.

Question 49 49 of 479 selected Nonlinear Functions M

The function f(t)=40,000(2)t790 gives the number of bacteria in a population t minutes after an initial observation. How much time, in minutes, does it take for the number of bacteria in the population to double?

  1. 2

  2. 790

  3. 1,580

  4. 40,000

Show Answer Correct Answer: B

Choice B is correct. It’s given that t minutes after an initial observation, the number of bacteria in a population is 40,000(2)t790. This expression consists of the initial number of bacteria, 40,000 , multiplied by the expression 2t790. The time, in minutes, it takes for the number of bacteria to double is the increase in the value of t that causes the expression 2t790 to double. Since the base is 2 , the expression 2t790 will double when the exponent increases by 1 . Since the exponent of this expression is t 790 , the exponent will increase by 1 when t increases by 790 . Therefore, the time, in minutes, it takes for the number of bacteria in the population to double is 790

Choice A is incorrect. This is the base of the exponent, not the time it takes for the number of bacteria in the population to double.

Choice C is incorrect. This is the number of minutes it takes for the population to double twice.

Choice D is incorrect. This is the number of bacteria that are initially observed, not the time it takes for the number of bacteria in the population to double.

Question 50 50 of 479 selected Nonlinear Functions E

  • The parabola opens upward.
  • The parabola passes through the following points:
    • (0 comma 0)
    • (3 comma negative 12)
    • (6 comma 0)

Scientists recorded data about the ocean water levels at a certain location over a period of 6 hours. The graph shown models the data, where y = 0 represents sea level. Which table gives values of x and their corresponding values of y based on the model?

Show Answer Correct Answer: C

Choice C is correct. Each point (x,y) on the graph represents an elapsed time x , in hours, and the corresponding ocean water level y , in feet, at a certain location based on the model. The graph shown passes through the points (0,0), (3,-12), and (6,0). Thus, the table in choice C gives the values of x and their corresponding values of y based on the model.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 51 51 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

|x+45|=48

What is the positive solution to the given equation?

  1. 3

  2. 48

  3. 93

  4. 96

Show Answer Correct Answer: A

Choice A is correct. The given absolute value equation can be rewritten as two linear equations: x + 45 = 48 and -(x+45)=48, or x + 45 = - 48 . Subtracting 45 from both sides of the equation x + 45 = 48 yields x = 3 . Subtracting 45 from both sides of the equation x + 45 = - 48 yields x = - 93 . Thus, the given equation has two possible solutions, 3 and - 93 . Therefore, the positive solution to the given equation is 3 .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 52 52 of 479 selected Nonlinear Functions E

P(t)=24.8(1.036)t

The function P gives the predicted population, in millions, of a certain country for the period from 1984 to 2018, where t is the number of years after 1984. According to the model, what is the best interpretation of the statement “P(8) is approximately equal to 32.91”?

  1. In 1984, the predicted population of this country was approximately 8 million.

  2. In 1984, the predicted population of this country was approximately 32.91 million.

  3. 8 years after 1984, the predicted population of this country was approximately 32.91 million.

  4. 32.91 years after 1984, the predicted population of this country was approximately 8 million.

Show Answer Correct Answer: C

Choice C is correct. The function P gives the predicted population, in millions, of a certain country for the period from 1984 to 2018, where t is the number of years after 1984. Since the value of P(8) is the value of P(t) when t = 8 , it follows that "P(8) is approximately equal to 32.91" means that the value of P(t) is approximately equal to 32.91 when t = 8 . Therefore, the best interpretation of the statement "P(8) is approximately equal to 32.91 " is that 8 years after 1984, the predicted population of this country was approximately 32.91 million.

Choice A is incorrect. In 1984, the predicted population of this country was 24.8 million, not approximately 8 million.

Choice B is incorrect. In 1984, the predicted population of this country was 24.8 million, not approximately 32.91 million.

Choice D is incorrect. 32.91 years after 1984, the predicted population of this country was 24.8(1.036)32.91 million, or approximately 79.42 million, not approximately 8 million.

Question 53 53 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

6r=7s+t

The given equation relates the variables r , s , and t . Which equation correctly expresses s in terms of r and t ?

  1. s=42r-t

  2. s=7(6r-t)

  3. s=67r-t

  4. s=6r-t7

Show Answer Correct Answer: D

Choice D is correct. Subtracting t from both sides of the given equation yields 6r-t=7s. Dividing both sides of this equation by 7 yields 6r-t7=s. Therefore, the equation s=6r-t7 correctly expresses s in terms of r and t .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 54 54 of 479 selected Nonlinear Functions M

A company has a newsletter. In January 2018, there were 1,300 customers subscribed to the newsletter. For the next 24 months after January 2018, the total number of customers subscribed to the newsletter each month was 7% greater than the total number subscribed the previous month. Which equation gives the total number of customers, c , subscribed to the company's newsletter m months after January 2018, where m24?

  1. c=1,300(0.07)m

  2. c=1,300(1.07)m

  3. c=1,300(1.7)m

  4. c=1,300(7)m

Show Answer Correct Answer: B

Choice B is correct. It's given that in January 2018, there were 1,300 customers subscribed to a company's newsletter and for the next 24 months after January 2018, the total number of customers subscribed to the newsletter each month was 7% greater than the total number subscribed the previous month. It follows that this situation can be represented by the equation c=a(1+r100)m, where c is the total number of customers subscribed to the company's newsletter m months after January 2018, a is the number of customers subscribed to the newsletter in January 2018, and the total number of customers subscribed to the newsletter each month was r% greater than the total number subscribed the previous month. Substituting 1,300 for a and 7 for r in this equation yields c=1,300(1+7100)m, or c=1,300(1.07)m.

Choice A is incorrect. This equation represents a situation where the total number of customers subscribed each month was 93% less, not 7% greater, than the total number subscribed the previous month.

Choice C is incorrect. This equation represents a situation where the total number of customers subscribed each month was 70%, not 7%, greater than the total number subscribed the previous month.

Choice D is incorrect. This equation represents a situation where the total number of customers subscribed each month was 600%, not 7%, greater than the total number subscribed the previous month.

Question 55 55 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

the fraction with numerator 1, and denominator x squared, plus 10 x, plus 25, end fraction, equals 4

If x is a solution to the given equation, which of the following is a possible value of x plus 5 ?

  1. one half

  2. five halves

  3. nine halves

  4. eleven halves

Show Answer Correct Answer: A

Choice A is correct. The given equation can be rewritten as the fraction 1 over, open parenthesis, x plus 5, close parenthesis, squared, end fraction, equals 4. Multiplying both sides of this equation by open parenthesis, x plus 5, close parenthesis, squared yields 1 equals, 4 times, open parenthesis, x plus 5, close parenthesis, squared. Dividing both sides of this equation by 4 yields one fourth equals, open parenthesis, x plus 5, close parenthesis, squared. Taking the square root of both sides of this equation yields one half equals, x plus 5 or negative one half equals, x plus 5. Therefore, a possible value of x plus 5 is one half.
Choices B, C, and D are incorrect and may result from computational or conceptual errors.

Question 56 56 of 479 selected Nonlinear Functions E

The function f is defined by f(x)=4+x. What is the value of f(144)?

  1. 0

  2. 16

  3. 40

  4. 76

Show Answer Correct Answer: B

Choice B is correct. The value of f(144) is the value of f(x) when x=144. It's given that the function f is defined by f(x)=4+x. Substituting 144 for x in this equation yields f(144)=4+144. Since the positive square root of 144 is 12 , it follows that this equation can be rewritten as f(144)=4+12, or f(144)=16. Therefore, the value of f(144) is 16 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the value of f(1,296), not f(144).

Choice D is incorrect. This is the value of f(5,184), not f(144).

Question 57 57 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

k-x=58-x

In the given equation, k is a constant. The equation has exactly one real solution. What is the minimum possible value of 4 k ?

Show Answer Correct Answer: 231

The correct answer is 231 . It's given that k-x=58-x. Squaring both sides of this equation yields k-x=(58-x)2, which is equivalent to the given equation if 58-x>0. It follows that if a solution to the equation k-x=(58-x)2 satisfies 58-x>0, then it's also a solution to the given equation; if not, it's extraneous. The equation k-x=(58-x)2 can be rewritten as k-x=3,364-116x+x2. Adding x to both sides of this equation yields k=x2-115x+3,364. Subtracting k from both sides of this equation yields 0=x2-115x+(3,364-k). The number of solutions to a quadratic equation in the form 0=ax2+bx+c, where a , b , and c are constants, can be determined by the value of the discriminant, b2-4ac. Substituting - 115 for b , 1 for a , and 3,364-k for c in b2-4ac yields (-115)2-4(1)(3,364-k), or 4 k - 231 . The equation 0=x2-115x+(3,364-k) has exactly one real solution if the discriminant is equal to zero, or 4 k - 231 = 0 . Subtracting 231 from both sides of this equation yields 4 k = 231 . Dividing both sides of this equation by 4 yields k=57.75. Therefore, if k=57.75, then the equation 0=x2-115x+(3,364-k) has exactly one real solution. Substituting 57.75 for k in this equation yields 0=x2-115x+(3,364-57.75), or 0=x2-115x+3,306.25, which is equivalent to 0=(x-57.5)2. Taking the square root of both sides of this equation yields 0=x-57.5. Adding 57.5 to both sides of this equation yields 57.5=x. To check whether this solution satisfies 58-x>0, the solution, 57.5, can be substituted for x in 58-x>0, which yields 58-57.5>0, or 0.5>0. Since 0.5 is greater than 0 , it follows that if k=57.75, or 4 k = 231 , then the given equation has exactly one real solution. If 4k<231, then the discriminant, 4 k - 231 , is negative and the given equation has no solutions. Therefore, the minimum possible value of 4 k is 231 .

Question 58 58 of 479 selected Nonlinear Functions H

f(x)=(x-1)(x+3)(x-2)

In the xy-plane, when the graph of the function f , where y=f(x), is shifted up 6 units, the resulting graph is defined by the function g . If the graph of y=g(x) crosses through the point (4,b), where b is a constant, what is the value of b ?

Show Answer Correct Answer: 48

The correct answer is 48 . It's given that in the xy-plane, when the graph of the function f , where y=f(x), is shifted up 6 units, the resulting graph is defined by the function g . Therefore, function g can be defined by the equation g(x)=f(x)+6. It's given that f(x)=(x-1)(x+3)(x-2). Substituting (x-1)(x+3)(x-2) for f(x) in the equation g(x)=f(x)+6 yields g(x)=(x-1)(x+3)(x-2)+6. For the point (4,b), the value of x is 4 . Substituting 4 for x in the equation g(x)=(x-1)(x+3)(x-2)+6 yields g(4)=(4-1)(4+3)(4-2)+6, or g(4)=48. It follows that the graph of y=g(x) crosses through the point (4,48). Therefore, the value of b is 48 .

Question 59 59 of 479 selected Nonlinear Functions E

The function g is defined by g(x)=8x+1. What is the value of g(3)?

  1. 5 8

  2. 25 8

  3. 5

  4. 25

Show Answer Correct Answer: C

Choice C is correct. It’s given that the function g is defined by g(x)=8x+1. Substituting 3 for x in the given function yields g(3)=8(3)+1, which is equivalent to g(3)=25, or g(3)=5. Therefore, the value of g(3) is 5 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the value of 8(3)+1, not 8(3)+1.

Question 60 60 of 479 selected Equivalent Expressions M

Which of the following expressions is equivalent to 8 x 10 - 8 x 9 + 88 x ?

  1. x(7x10-7x9+87x)

  2. x(810-89+88)

  3. 8x(x10-x9+11x)

  4. 8x(x9-x8+11)

Show Answer Correct Answer: D

Choice D is correct. Since 8 x is a common factor of each term in the given expression, the expression can be rewritten as 8x(x9-x8+11).

Choice A is incorrect. This expression is equivalent to 7 x 11 - 7 x 10 + 87 x 2 .

Choice B is incorrect. This expression is equivalent to 810x-89x+88x.

Choice C is incorrect. This expression is equivalent to 8 x 11 - 8 x 10 + 88 x 2 .

Question 61 61 of 479 selected Nonlinear Functions M

A rectangular volleyball court has an area of 162 square meters. If the length of the court is twice the width, what is the width of the court, in meters?

  1. 9

  2. 18

  3. 27

  4. 54

Show Answer Correct Answer: A

Choice A is correct. It’s given that the volleyball court is rectangular and has an area of 162 square meters. The formula for the area of a rectangle is A equals l w , where A is the area, l is the length, and w is the width of the rectangle. It’s also given that the length of the volleyball court is twice the width, thus l equals 2 w. Substituting the given value into the formula for the area of a rectangle and using the relationship between length and width for this rectangle yields 162 equals, 2 w times w . This equation can be rewritten as 162 equals, 2 w squared. Dividing both sides of this equation by 2 yields 81 equals, w squared. Taking the square root of both sides of this equation yields plus or minus 9 equals w. Since the width of a rectangle is a positive number, the width of the volleyball court is 9 meters.

Choice B is incorrect because this is the length of the rectangle. Choice C is incorrect because this is the result of using 162 as the perimeter rather than the area. Choice D is incorrect because this is the result of calculating w in the equation 162 equals, 2 w plus w instead of 162 equals, 2 w times w.

Question 62 62 of 479 selected Nonlinear Functions H

A machine launches a softball from ground level. The softball reaches a maximum height of 51.84 meters above the ground at 1.8 seconds and hits the ground at 3.6 seconds. Which equation represents the height above ground h , in meters, of the softball t seconds after it is launched?

  1. h = - t 2 + 3.6

  2. h = - t 2 + 51.84

  3. h = - 16 ( t - 1.8 ) 2 - 3.6

  4. h = - 16 ( t - 1.8 ) 2 + 51.84

Show Answer Correct Answer: D

Choice D is correct. An equation representing the height above ground h , in meters, of a softball t seconds after it is launched by a machine from ground level can be written in the form h=-a(t-b)2+c, where a , b , and c are positive constants. In this equation, b represents the time, in seconds, at which the softball reaches its maximum height of c meters above the ground. It's given that this softball reaches a maximum height of 51.84 meters above the ground at 1.8 seconds; therefore, b=1.8 and c = 51.84 . Substituting 1.8 for b and 51.84 for c in the equation h=-a(t-b)2+c yields h=-a(t-1.8)2+51.84. It's also given that this softball hits the ground at 3.6 seconds; therefore, h=0 when t=3.6. Substituting 0 for h and 3.6 for t in the equation h=-a(t-1.8)2+51.84 yields 0=-a(3.6-1.8)2+51.84, which is equivalent to 0=-a(1.8)2+51.84, or 0=-3.24a+51.84. Adding 3.24 a to both sides of this equation yields 3.24a=51.84. Dividing both sides of this equation by 3.24 yields a=16. Substituting 16 for a in the equation h=-a(t-1.8)2+51.84 yields h=-16(t-1.8)2+51.84. Therefore, h=-16(t-1.8)2+51.84 represents the height above ground h , in meters, of this softball t seconds after it is launched.

Choice A is incorrect. This equation represents a situation where the maximum height is 3.6 meters above the ground at 0 seconds, not 51.84 meters above the ground at 1.8 seconds.

Choice B is incorrect. This equation represents a situation where the maximum height is 51.84 meters above the ground at 0 seconds, not 1.8 seconds.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 63 63 of 479 selected Nonlinear Functions H

The function f is defined by f(x)=ax+b, where a and b are constants. In the x y -plane, the graph of y=f(x) has an x -intercept at (2,0) and a y -intercept at (0,-323). What is the value of b ?

Show Answer Correct Answer: -324

The correct answer is -324. It's given that the function f is defined by f(x)=ax+b, where a and b are constants. It's also given that the graph of y=f(x) has a y-intercept at (0,-323). It follows that f(0)=-323. Substituting 0 for x and -323 for f(x) in f(x)=ax+b yields -323=a0+b, or -323=1+b. Subtracting 1 from each side of this equation yields -324=b. Therefore, the value of b is -324.

Question 64 64 of 479 selected Nonlinear Functions M

For the exponential function f , the value of f(0) is c , where c is a constant. Of the following equations that define the function f , which equation shows the value of c as the coefficient or the base?

  1. f(x)=22(1.5)x+1

  2. f(x)=33(1.5)x

  3. f(x)=49.5(1.5)x-1

  4. f(x)=74.25(1.5)x-2

Show Answer Correct Answer: B

Choice B is correct. Each of the given choices is an equation of the form f(x)=a(b)x-k, where a , b , and k are constants. For an equation of this form, the coefficient, a , is equal to the value of the function when the exponent is equal to 0 , or when x = k . It follows that in the equation f(x)=33(1.5)x, the coefficient, 33 , is equal to the value of f(0). Substituting 0 for x in this equation yields f(0)=33(1.5)0, which is equivalent to f(0)=33(1), or f(0)=33. Thus, the value of c is 33 and the equation f(x)=33(1.5)x shows the value of c as the coefficient.

Choice A is incorrect. This equation shows the value of f(-1), not f(0), as the coefficient.

Choice C is incorrect. This equation shows the value of f(1), not f(0), as the coefficient.

Choice D is incorrect. This equation shows the value of f(2), not f(0), as the coefficient.

Question 65 65 of 479 selected Nonlinear Functions M

S of n, equals, 38,000 times a, to the n power

The function S above models the annual salary, in dollars, of an employee n years after starting a job, where a is a constant. If the employee’s salary increases by 4% each year, what is the value of a ?

  1. 0.04

  2. 0.4

  3. 1.04

  4. 1.4

Show Answer Correct Answer: C

Choice C is correct. A model for a quantity S that increases by a certain percentage per time period n is an exponential function in the form S of n equals, I times, open parenthesis, 1 plus r over 100, close parenthesis, to the n power, where I is the initial value at time n equals 0 for r% annual increase. It’s given that the annual increase in an employee’s salary is 4%, so r equals 4. The initial value can be found by substituting 0 for n in the given function, which yields S of 0 equals 38,000. Therefore, I equals 38,000. Substituting these values for r and I into the form of the exponential function S of n equals, I times, open parenthesis, 1 plus r over 100, close parenthesis, to the n power yields S of n equals, 38,000 times, open parenthesis, 1 plus 4 over 100, close parenthesis, to the n power, or S of n equals, 38,000 times, 1 point 0 4 to the n power. Therefore, the value of a in the given function is 1.04.

Choices A, B, and D are incorrect and may result from incorrectly representing the annual increase in the exponential function.

 

Question 66 66 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

During a 5-second time interval, the average acceleration a, in meters per second squared, of an object with an initial velocity of 12 meters per second is defined by the equation a equals, the fraction with numerator v subscript f, end subscript, minus 12, and denominator 5, where vf is the final velocity of the object in meters per second. If the equation is rewritten in the form vf = xa + y, where x and y are constants, what is the value of x ?

Show Answer

The correct answer is 5. The given equation can be rewritten in the form v subscript f equals, x a, plus y, like so:

a, equals, the fraction with numerator v subscript f, end subscript, minus 12, and denominator 5

v subscript f, end subscript, minus 12, equals 5 a

v subscript f equals, 5 a, plus 12

It follows that the value of x is 5 and the value of y is 12.

Question 67 67 of 479 selected Equivalent Expressions M

The expression 90 y 5 - 54 y 4 is equivalent to ry4(15y-9), where r is a constant. What is the value of r ?

Show Answer Correct Answer: 6

The correct answer is 6 . Applying the distributive property to the expression ry4(15y-9) yields 15ry5-9ry4. Since 90y5-54y4 is equivalent to ry4(15y-9), it follows that 90y5-54y4 is also equivalent to 15ry5-9ry4. Since these expressions are equivalent, it follows that corresponding coefficients are equivalent. Therefore, 90 = 15 r and -54 = - 9 r . Solving either of these equations for r will yield the value of r . Dividing both sides of 90 = 15 r by 15 yields 6 = r . Therefore, the value of r is 6 .  

Question 68 68 of 479 selected Nonlinear Functions H

f of x equals, negative 500, x squared, plus 25,000 x

The revenue f of x , in dollars, that a company receives from sales of a product is given by the function f above, where x is the unit price, in dollars, of the product. The graph of y equals f of x in the xy-plane intersects the x-axis at 0 and a. What does a represent?

 

  1. The revenue, in dollars, when the unit price of the product is $0

  2. The unit price, in dollars, of the product that will result in maximum revenue

  3. The unit price, in dollars, of the product that will result in a revenue of $0

  4. The maximum revenue, in dollars, that the company can make

Show Answer Correct Answer: C

Choice C is correct. By definition, the y-value when a function intersects the x-axis is 0. It’s given that the graph of the function intersects the x-axis at 0 and a, that x is the unit price, in dollars, of a product, and that f of x, where y equals f of x, is the revenue, in dollars, that a company receives from the sales of the product. Since the value of a occurs when y equals 0, a is the unit price, in dollars, of the product that will result in a revenue of $0.

Choice A is incorrect. The revenue, in dollars, when the unit price of the product is $0 is represented by f of x, when x equals 0. Choice B is incorrect. The unit price, in dollars, of the product that will result in maximum revenue is represented by the x-coordinate of the maximum of f. Choice D is incorrect. The maximum revenue, in dollars, that the company can make is represented by the y-coordinate of the maximum of f.

 

Question 69 69 of 479 selected Nonlinear Functions M

f(x)=3,000(0.75)x

A conservation scientist implemented a program to reduce the population of a certain species in an area. The given function estimates this species' population x years after 2008, where x8. Which of the following is the best interpretation of 3,000 in this context?

  1. The estimated percent decrease in the population for this species and area every 8 years after 2008

  2. The estimated percent decrease in the population for this species and area each year after 2008

  3. The estimated population for this species and area 8 years after 2008

  4. The estimated initial population for this species and area in 2008

Show Answer Correct Answer: D

Choice D is correct. Substituting 0 for x in the given equation yields f(0)=3,000(0.75)0, which is equivalent to f(0)=3,000(1), or f(0)=3,000. It’s given that the function estimates the species’ population x years after 2008, so it follows that the estimated population of the species is 3,000 in 2008. Therefore, the best interpretation of 3,000 in this context is the estimated initial population for this species and area in 2008.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect. The estimated percent decrease in the population for this species and area each year after 2008 is 25%, not 3,000 .

Choice C is incorrect. The estimated population for this species and area 8 years after 2008 is 3,000(0.75)8, or approximately 300 , not 3,000 .

Question 70 70 of 479 selected Equivalent Expressions H

the fraction with numerator 2, and denominator x minus 2, end fraction, plus, the fraction with numerator 3, and denominator x plus 5, end fraction, equals, the fraction with numerator r x plus t, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction

The equation above is true for all x greater than 2, where r and t are positive constants. What is the value of rt ?

  1. negative 20

  2. 15

  3. 20

  4. 60

Show Answer Correct Answer: C

Choice C is correct. To express the sum of the two rational expressions on the left-hand side of the equation as the single rational expression on the right-hand side of the equation, the expressions on the left-hand side must have the same denominator. Multiplying the first expression by the fraction with numerator x plus 5, and denominator x minus 5, end fraction results in the fraction with numerator 2 times, open parenthesis, x plus 5, close parenthesis, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction, and multiplying the second expression by the fraction with numerator x minus 2, and denominator x minus 2, end fraction results in the fraction with numerator 3 times, open parenthesis, x minus 2, close parenthesis, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction, so the given equation can be rewritten as the fraction with numerator 2 times, open parenthesis, x plus 5, close parenthesis, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction, plus, the fraction with numerator 3 times, open parenthesis, x minus 2, close parenthesis, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction, equals, the fraction with numerator r x plus t, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction, or the fraction with numerator 2 x plus 10, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction, plus, the fraction with numerator 3 x minus 6, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction, equals, the fraction with numerator r x plus t, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction. Since the two rational expressions on the left-hand side of the equation have the same denominator as the rational expression on the right-hand side of the equation, it follows that open parenthesis, 2 x plus 10, close parenthesis, plus, open parenthesis, 3 x minus 6, close parenthesis, equals, r x plus t. Combining like terms on the left-hand side yields 5 x plus 4, equals, r x plus t, so it follows that r equals 5 and t equals 4. Therefore, the value of r t is 5 times 4, which equals 20.

Choice A is incorrect and may result from an error when determining the sign of either r or t. Choice B is incorrect and may result from not distributing the 2 and 3 to their respective terms in the fraction with numerator 2 times, open parenthesis, x plus 5, close parenthesis, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction, plus, the fraction with numerator 3 times, open parenthesis, x minus 2, close parenthesis, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction, equals, the fraction with numerator r x plus t, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction. Choice D is incorrect and may result from a calculation error.

 

Question 71 71 of 479 selected Equivalent Expressions M

Which of the following is an equivalent form of open parenthesis, 1 point 5 x minus 2 point 4, close parenthesis, squared, minus, open parenthesis, 5 point 2, x squared, minus 6 point 4, close parenthesis ?

  1. negative 2 point 2, x squared, plus 1 point 6
  2. negative 2 point 2, x squared, plus 11 point 2
  3. negative 2 point 95, x squared, minus 7 point 2 x, plus 12 point 16
  4. negative 2 point 95, x squared, minus 7 point 2 x, plus 0 point 64
Show Answer Correct Answer: C

Choice C is correct. The first expression open parenthesis, 1 point 5 x minus 2 point 4, close parenthesis, squared can be rewritten as open parenthesis, 1 point 5 x minus 2 point 4, close parenthesis, times, open parenthesis, 1 point 5 x minus 2 point 4, close parenthesis . Applying the distributive property to this product yields open parenthesis, 2 point 2 5, x squared, minus 3 point 6 x, minus 3 point 6 x, plus 5 point 7 6, close parenthesis, minus, open parenthesis, 5 point 2, x squared, minus 6 point 4, close parenthesis . This difference can be rewritten as open parenthesis, 2 point 2 5, x squared, minus 3 point 6 x, minus 3 point 6 x, plus 5 point 7 6, close parenthesis, plus, negative 1 times, open parenthesis, 5 point 2 x squared, minus 6 point 4, close parenthesis . Distributing the factor of negative 1 through the second expression yields 2 point 2 5 x, squared, minus 3 point 6 x, minus 3 point 6 x, plus 5 point 7 6, minus 5 point 2, x squared, plus 6 point 4 . Regrouping like terms, the expression becomes open parenthesis, 2 point 2 5, x squared, minus 5 point 2, x squared, close parenthesis, plus, open parenthesis, negative 3 point 6 x minus 3 point 6 x, close parenthesis, plus, open parenthesis, 5 point 7 6 plus 6 point 4, close parenthesis . Combining like terms yields negative 2 point 9 5, x squared, minus 7 point 2 x, plus 12 point 1 6 .

Choices A, B, and D are incorrect and likely result from errors made when applying the distributive property or combining the resulting like terms.

Question 72 72 of 479 selected Nonlinear Functions H
Growth of a Culture of Bacteria
Day Number of bacteria per
milliliter at end of day
1 2 point 5 times 10 to the power 5
2 5 point 0 times 10 to the power 5
3 1 point 0 times 10 to the power 6

A culture of bacteria is growing at an exponential rate, as shown in the table above. At this rate, on which day would the number of bacteria per milliliter reach 5 point 1 2, times 10, to the power 8?

  1. Day 5

  2. Day 9

  3. Day 11

  4. Day 12

Show Answer Correct Answer: D

Choice D is correct. The number of bacteria per milliliter is doubling each day. For example, from day 1 to day 2, the number of bacteria increased from 2.5 × 105 to 5.0 × 105. At the end of day 3 there are 106 bacteria per milliliter. At the end of day 4, there will be 106 × 2 bacteria per milliliter. At the end of day 5, there will be (106 × 2) × 2, or 106 × (22) bacteria per milliliter, and so on. At the end of day d, the number of bacteria will be 106 × (2d – 3). If the number of bacteria per milliliter will reach 5.12 × 108 at the end of day d, then the equation 10 to the power 6, end power, times 2 to the power d minus 3, end power, equals 5 point 1 2, times 10 to the power 8 must hold. Since 5.12 × 108 can be rewritten as 512 × 106,  the equation is equivalent to 2 to the power d minus 3, end power, equals 512. Rewriting 512 as 29 gives d – 3 = 9, so d = 12. The number of bacteria per milliliter would reach 5.12 × 108 at the end of day 12.

Choices A, B, and C are incorrect. Given the growth rate of the bacteria, the number of bacteria will not reach 5.12 × 108 per milliliter by the end of any of these days.

Question 73 73 of 479 selected Equivalent Expressions E

Which expression is equivalent to (8yz)(y)(7z)?

  1. 56 y 2 z 2

  2. 56 y 2 z

  3. 56 y z

  4. 16 y z

Show Answer Correct Answer: A

Choice A is correct. The given expression can be rewritten as (8·7)(y·y)(z·z), which is equivalent to (56)(y2)(z2), or 56 y 2 z 2 .

Choice B is incorrect. This expression is equivalent to (8yz)(y)(7).

Choice C is incorrect. This expression is equivalent to (8z)(y)(7).

Choice D is incorrect and may result from conceptual or calculation errors.

Question 74 74 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

2 x 2 - 8 x - 7 = 0

One solution to the given equation can be written as 8-k4, where k is a constant. What is the value of k ?

Show Answer Correct Answer: 120

The correct answer is 120 . The solutions to a quadratic equation of the form a x 2 + b x + c = 0 can be calculated using the quadratic formula and are given by x=-b±b2-4ac2a. The given equation is in the form a x 2 + b x + c = 0 , where a = 2 , b = - 8 , and c = - 7 . It follows that the solutions to the given equation are x=8±(-8)2-4(2)(-7)2(2), which is equivalent to x=8±64+564, or x=8±1204. It's given that one solution to the equation 2 x 2 - 8 x - 7 = 0 can be written as 8-k4. The solution 8-1204 is in this form. Therefore, the value of k is 120 .

Question 75 75 of 479 selected Nonlinear Functions M

D equals, 5,640 times, open parenthesis, 1 point 9, close parenthesis, to the t power

The equation above estimates the global data traffic D, in terabytes, for the year that is t years after 2010. What is the best interpretation of the number 5,640 in this context?

  1. The estimated amount of increase of data traffic, in terabytes, each year

  2. The estimated percent increase in the data traffic, in terabytes, each year

  3. The estimated data traffic, in terabytes, for the year that is t years after 2010

  4. The estimated data traffic, in terabytes, in 2010

Show Answer Correct Answer: D

Choice D is correct. Since t represents the number of years after 2010, the estimated data traffic, in terabytes, in 2010 can be calculated using the given equation when t equals 0. Substituting 0 for t in the given equation yields D equals, 5,640 times, open parenthesis, 1 point 9, close parenthesis, to the 0 power, or 5,640 times 1, equals 5,640. Thus, 5,640 represents the estimated data traffic, in terabytes, in 2010.

Choice A is incorrect. Since the equation is exponential, the amount of increase of data traffic each year isn’t constant. Choice B is incorrect. According to the equation, the percent increase in data traffic each year is 90%. Choice C is incorrect. The estimated data traffic, in terabytes, for the year that is t years after 2010 is represented by D, not the number 5,640.

 

Question 76 76 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

y = 76
y = x 2 - 5

The graphs of the given equations in the xy-plane intersect at the point (x,y). What is a possible value of x ?

  1. - 76 5

  2. -9

  3. 5

  4. 76

Show Answer Correct Answer: B

Choice B is correct. Since the point (x,y) is an intersection point of the graphs of the given equations in the xy-plane, the pair (x,y) should satisfy both equations, and thus is a solution of the given system. According to the first equation, y = 76 . Substituting 76 in place of y in the second equation yields x 2 - 5 = 76 . Adding 5 to both sides of this equation yields x 2 = 81 . Taking the square root of both sides of this equation yields two solutions: x = 9 and x = -9 . Of these two solutions, only -9 is given as a choice.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the value of coordinate y , rather than x , of the intersection point (x,y).

Question 77 77 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

7m=5(n+p)

The given equation relates the positive numbers m , n , and p . Which equation correctly gives n in terms of m and p ?

  1. n=5p7m

  2. n=7m5-p

  3. n=5(7m)+p

  4. n=7m-5-p

Show Answer Correct Answer: B

Choice B is correct. It's given that the equation 7m=5(n+p) relates the positive numbers m , n , and p . Dividing both sides of the given equation by 5 yields 7m5=n+p. Subtracting p from both sides of this equation yields 7m5-p=n, or n=7m5-p. It follows that the equation n=7m5-p correctly gives n in terms of m and p .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 78 78 of 479 selected Equivalent Expressions H

70n5(70n6)2

For what value of x is the given expression equivalent to (70n)30x, where n>1?

Show Answer Correct Answer: .0177, .0178, 4/225

The correct answer is 4 225 . An expression of the form ak, where k is an integer greater than 1 and a0, is equivalent to a1k. Therefore, the given expression, where n>1, is equivalent to (70n)15((70n)16)2. Applying properties of exponents, this expression can be rewritten as (70n)15(70n)16·2, or (70n)15(70n)13, which can be rewritten as (70n)15+13, or (70n)815. It's given that the expression 70n5(70n6)2 is equivalent to (70n)30x, where n>1. It follows that (70n)815 is equivalent to (70n)30x. Therefore, 815=30x. Dividing both sides of this equation by 30 yields 8450=x, or 4225=x. Thus, the value of x for which the given expression is equivalent to (70n)30x, where n>1, is 4 225 . Note that 4/225, .0177, .0178, 0.017, and 0.018 are examples of ways to enter a correct answer.

Question 79 79 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

2 x squared, minus 2, equals 2 x plus 3

Which of the following is a solution to the equation above?

  1. 2

  2. 1 minus the square root of 11

  3. one half plus the square root of 11

  4. the fraction with numerator 1 plus the square root of 11, and denominator 2

Show Answer Correct Answer: D

Choice D is correct. A quadratic equation in the form a, x squared, plus b x, plus c, equals 0, where a, b, and c are constants, can be solved using the quadratic formula: x equals, the fraction with numerator negative b plus or minus the square root of b squared, minus 4 a c, end root, and denominator 2 a, end fraction. Subtracting 2 x plus 3 from both sides of the given equation yields 2 x squared, minus 2 x, minus 5, equals 0. Applying the quadratic formula, where a, equals 2, b equals negative 2, and c equals negative 5, yields x equals the fraction with numerator negative, open parenthesis, negative 2, close parenthesis, plus or minus the square root of, open parenthesis, negative 2, close parenthesis, squared, minus 4 times 2, times negative 5, end root, and denominator 2 times 2, end fraction. This can be rewritten as x equals, the fraction with numerator 2 plus or minus the square root of 44, end root, and denominator 4 . Since the square root of 44 equals, the square root of 2 squared, times 11, end root, or 2 times the square root of 11, the equation can be rewritten as x equals, the fraction with numerator 2 plus or minus 2 times the square root of 11, end root, and denominator 4. Dividing 2 from both the numerator and denominator yields the fraction with numerator 1 plus the square root of 11, end root, and denominator 2 or the fraction with numerator 1 minus the square root of 11, end root, and denominator 2. Of these two solutions, only the fraction with numerator 1 plus the square root of 11, end root, and denominator 2 is present among the choices. Thus, the correct choice is D.

Choice A is incorrect and may result from a computational or conceptual error. Choice B is incorrect and may result from using x equals, the fraction with numerator negative b plus or minus the square root of b squared, minus 4 a c, end root, and denominator a, end fraction instead of x equals, the fraction with numerator negative b, plus or minus the square root of b squared, minus 4 a c, end root, and denominator 2 a, end fraction as the quadratic formula. Choice C is incorrect and may result from rewriting the square root of 44 as 4 times the square root of 11 instead of 2 times the square root of 11.

 

Question 80 80 of 479 selected Nonlinear Functions H

f(x)=ax2+4x+c

In the given quadratic function, a and c are constants. The graph of y=f(x) in the xy-plane is a parabola that opens upward and has a vertex at the point (h,k), where h and k are constants. If k<0 and f(-9)=f(3), which of the following must be true?

  1. c<0
  2. a1
  1. I only

  2. II only

  3. I and II

  4. Neither I nor II

Show Answer Correct Answer: D

Choice D is correct. It's given that the graph of y=f(x) in the xy-plane is a parabola with vertex (h,k). If f(-9)=f(3), then for the graph of y=f(x), the point with an x-coordinate of -9 and the point with an x-coordinate of 3 have the same y-coordinate. In the xy-plane, a parabola is a symmetric graph such that when two points have the same y-coordinate, these points are equidistant from the vertex, and the x-coordinate of the vertex is halfway between the x-coordinates of these two points. Therefore, for the graph of y=f(x), the points with x-coordinates -9 and 3 are equidistant from the vertex, (h,k), and h is halfway between -9 and 3 . The value that is halfway between -9 and 3 is -9+32, or -3 . Therefore, h = -3 . The equation defining f can also be written in vertex form, f(x)=a(x-h)2+k. Substituting -3 for h in this equation yields f(x)=a(x-(-3))2+k, or f(x)=a(x+3)2+k. This equation is equivalent to f(x)=a(x2+6x+9)+k, or f(x)=ax2+6ax+9a+k. Since f(x)=ax2+4x+c, it follows that 6 a = 4 and 9 a + k = c . Dividing both sides of the equation 6 a = 4 by 6 yields a=46, or a = 2 3 . Since 23<1, it's not true that a1. Therefore, statement II isn't true. Substituting 2 3 for a in the equation 9 a + k = c yields 9(23)+k=c, or 6+k=c. Subtracting 6 from both sides of this equation yields k = c - 6 . If k<0, then c-6<0, or c<6. Since c could be any value less than 6 , it's not necessarily true that c<0. Therefore, statement I isn't necessarily true. Thus, neither I nor II must be true.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 81 81 of 479 selected Nonlinear Functions E

  • The parabola opens upward.
  • The vertex is at point (7 comma 0).
  • The parabola passes through the following points:
    • (6 comma 3)
    • (7 comma 0)
    • (8 comma 3)

The x-intercept of the graph shown is (x,0). What is the value of x ?

Show Answer Correct Answer: 7

The correct answer is 7 . It’s given that the x-intercept of the graph shown is (x,0). The graph passes through the point (7,0). Therefore, the value of x is 7

Question 82 82 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

x 2 = -841

How many distinct real solutions does the given equation have?

  1. Exactly one

  2. Exactly two

  3. Infinitely many

  4. Zero

Show Answer Correct Answer: D

Choice D is correct. Since the square of a real number is never negative, the given equation isn't true for any real value of x . Therefore, the given equation has zero distinct real solutions.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 83 83 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

In the xy-plane, a line with equation 2 y = c for some constant c intersects a parabola at exactly one point. If the parabola has equation y = - 2 x 2 + 9 x , what is the value of c ?

Show Answer Correct Answer: 20.25, 81/4

The correct answer is 814. The given linear equation is 2y=c. Dividing both sides of this equation by 2 yields y=c2. Substituting c2 for y in the equation of the parabola yields c2=-2x2+9x. Adding 2x2 and - 9 x to both sides of this equation yields 2x2-9x+c2=0. Since it’s given that the line and the parabola intersect at exactly one point, the equation 2x2-9x+c2=0 must have exactly one solution. An equation of the form Ax2+Bx+C=0, where A , B , and C are constants, has exactly one solution when the discriminant, B2-4AC, is equal to 0 . In the equation 2x2-9x+c2=0, where A=2, B=-9, and C=c2, the discriminant is (-9)2-4(2)(c2). Setting the discriminant equal to 0 yields (-9)2-4(2)(c2)=0, or 81-4c=0. Adding 4 c to both sides of this equation yields 81=4c. Dividing both sides of this equation by 4 yields c=814. Note that 81/4 and 20.25 are examples of ways to enter a correct answer.

Question 84 84 of 479 selected Nonlinear Functions M

  • In quadrant 3:
    • The curve rises sharply to touch the x axis at point (negative 5 comma 0).
    • The curve falls gradually to a relative minimum at point (negative 2 comma negative 11).
    • The curve rises gradually to cross both axes at the origin.
  • In quadrant 1:
    • The curve rises gradually to a relative maximum at point (2.5 comma 21).
    • The curve falls sharply to cross the x axis at point (4 comma 0).
  • In quadrant 4 the curve falls sharply.

Which of the following could be the equation of the graph shown in the xy-plane?

  1. y=-110x(x-4)(x+5)

  2. y=-110x(x-4)(x+5)2

  3. y=-110x(x-5)(x+4)

  4. y=-110x(x-5)2(x+4)

Show Answer Correct Answer: B

Choice B is correct. Each of the given choices is an equation of the form y=-110x(x-a)m(x+b)n, where a , b , m , and n are positive constants. In the xy-plane, the graph of an equation of this form has x-intercepts at x = 0 , x = a , and x = - b . The graph shown has x-intercepts at x = 0 , x = 4 , and x = -5 . Therefore, a = 4 and b = 5 . Of the given choices, only choices A and B have a = 4 and b = 5 . For an equation in the form y=-110x(x-a)m(x+b)n, if all values of x that are less than - b or greater than a correspond to negative y-values, then the sum of all the exponents of the factors on the right-hand side of the equation is even. In the graph shown, all values of x less than -5 or greater than 4 correspond to negative y-values. Therefore, the sum of all the exponents of the factors on the right-hand side of the equation y=-110x(x-4)m(x+5)n must be even. For choice A, the sum of these exponents is 1+1+1, or 3 , which is odd. For choice B, the sum of these exponents is 1+1+2, or 4 , which is even. Therefore, y=-110x(x-4)(x+5)2 could be the equation of the graph shown.

Choice A is incorrect. For the graph of this equation, all values of x less than -5 correspond to positive, not negative, y-values.

Choice C is incorrect. The graph of this equation has x-intercepts at x = -4 , x = 0 , and x = 5 , rather than x-intercepts at x = -5 , x = 0 , and x = 4 .

Choice D is incorrect. The graph of this equation has x-intercepts at x = -4 , x = 0 , and x = 5 , rather than x-intercepts at x = -5 , x = 0 , and x = 4 .

Question 85 85 of 479 selected Nonlinear Functions H

P(t)=260(1.04)(64)t

The function P models the population, in thousands, of a certain city t years after 2003 . According to the model, the population is predicted to increase by 4% every n months. What is the value of n ?

  1. 8

  2. 12

  3. 18

  4. 72

Show Answer Correct Answer: A

Choice A is correct. It’s given that the function P models the population, in thousands, of a certain city t years after 2003. The value of the base of the given exponential function, 1.04 , corresponds to an increase of 4% for every increase of 1 in the exponent, (64)t. If the exponent is equal to 0 , then (64)t=0. Multiplying both sides of this equation by (46) yields t = 0 . If the exponent is equal to 1 , then (64)t=1. Multiplying both sides of this equation by (46) yields t=46, or t = 2 3 . Therefore, the population is predicted to increase by 4% every 2 3 of a year. It’s given that the population is predicted to increase by 4% every n months. Since there are 12 months in a year, 2 3 of a year is equivalent to (23)(12), or 8 , months. Therefore, the value of n is 8 .

Choice B is incorrect. This is the number of months in which the population is predicted to increase by 4% according to the model P(t)=260(1.04)t, not P(t)=260(1.04)(64)t.

Choice C is incorrect. This is the number of months in which the population is predicted to increase by 4% according to the model P(t)=260(1.04)(46)t, not P(t)=260(1.04)(64)t.

Choice D is incorrect. This is the number of months in which the population is predicted to increase by 4% according to the model P(t)=260(1.04)(16)t, not P(t)=260(1.04)(64)t.

Question 86 86 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

y=5x+4

y=5x2+4

Which ordered pair (x,y) is a solution to the given system of equations?

  1. (0,0)

  2. (0,4)

  3. (8,44)

  4. (8,84)

Show Answer Correct Answer: B

Choice B is correct. The second equation in the given system is y=5x2+4. Substituting 5x2+4 for y in the first equation of the given system yields 5x2+4=5x+4. Subtracting 4 from both sides of this equation yields 5x2=5x. Subtracting 5x from both sides of this equation yields 5x2-5x=0. Factoring out a common factor of 5x from the left-hand side of this equation yields 5x(x-1)=0. It follows that 5x=0 or x-1=0. Dividing both sides of the equation 5x=0 by 5 yields x=0. Substituting 0 for x in the equation y=5x+4 yields y=5(0)+4, or y=4. Therefore, a solution to the given system of equations is the ordered pair (0,4).

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 87 87 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

(d-30)(d+30)-7=-7

What is a solution to the given equation?

Show Answer Correct Answer: 30, -30

The correct answer is either -30 or 30 . Adding 7 to each side of the given equation yields (d-30)(d+30)=0. Since a product of two factors is equal to 0 if and only if at least one of the factors is 0 , either d - 30 = 0 or d + 30 = 0 . Adding 30 to each side of the equation d - 30 = 0 yields d = 30 . Subtracting 30 from each side of the equation d + 30 = 0 yields d = -30 . Therefore, the solutions to the given equation are -30 and 30 . Note that -30 and 30 are examples of ways to enter a correct answer.

Question 88 88 of 479 selected Nonlinear Functions H

f(x)= 4 x 2 + 64 x + 262

The function g is defined by g(x)=f(x+5). For what value of x does g(x) reach its minimum?

  1. -13

  2. -8

  3. -5

  4. -3

Show Answer Correct Answer: A

Choice A is correct. It's given that g(x)=f(x+5). Since f(x)=4x2+64x+262, it follows that f(x+5)=4(x+5)2+64(x+5)+262. Expanding the quantity (x+5)2 in this equation yields f(x+5)=4(x2+10x+25)+64(x+5)+262. Distributing the 4 and the 64 yields f(x+5)=4x2+40x+100+64x+320+262. Combining like terms yields f(x+5)=4x2+104x+682. Therefore, g(x)=4x2+104x+682. For a quadratic function defined by an equation of the form g(x)=a(x-h)2+k, where a , h , and k are constants and a is positive, g(x) reaches its minimum, k , when the value of x is h . The equation g(x)=4x2+104x+682 can be rewritten in this form by completing the square. This equation is equivalent to g(x)=4(x2+26)+682, or g(x)=4(x2+26x+169-169)+682. This equation can be rewritten as g(x)=4((x+13)2-169)+682, or g(x)=4(x+13)2-4(169)+682, which is equivalent to g(x)=4(x+13)2+6. This equation is in the form g(x)=a(x-h)2+k, where a = 4 , h = -13 , and k = 6 . Therefore, g(x) reaches its minimum when the value of x is -13 .

Choice B is incorrect. This is the value of x for which f(x), rather than g(x), reaches its minimum.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the value of x for which f(x-5), rather than f(x+5), reaches its minimum.

Question 89 89 of 479 selected Nonlinear Functions M

  • In quadrant 2 the curve falls sharply to cross the x axis at point (negative 1 comma 0).
  • In quadrant 3 the curve falls sharply to cross the y axis at point (0 comma negative 5.5).
  • In quadrant 4:
    • The curve falls gradually to a relative minimum at point (1 comma negative 7).
    • The curve rises sharply to cross the x axis at point (4 comma 0).
  • In quadrant 1:
    • The curve rises sharply to a relative maximum at point (5.5 comma 3).
    • The curve falls sharply to cross the x axis at point (7 comma 0).
  • In quadrant 4 the curve falls sharply. 

The graph of y=f(x) is shown, where the function f is defined by f(x)=ax3+bx2+cx+d and a , b , c , and d are constants. For how many values of x does f(x)=0?

  1. One

  2. Two

  3. Three

  4. Four

Show Answer Correct Answer: C

Choice C is correct. If a value of x  satisfies f(x)=0, the graph of y=f(x) will contain a point (x,0) and thus touch the x-axis. Since there are 3 points at which this graph touches the x-axis, there are 3 values of x for which f(x)=0.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 90 90 of 479 selected Equivalent Expressions E

Which expression is equivalent to 11x3-5x3?

  1. 16 x 3

  2. 6 x 3

  3. 6 x 6

  4. 16 x 6

Show Answer Correct Answer: B

Choice B is correct. The given expression can be rewritten as 11x3+(-5)x3. Since the two terms of this expression are both constant multiples of x3, they are like terms and can, therefore, be combined through addition. Adding like terms in the expression 11x3+(-5)x3 yields 6x3.

Choice A is incorrect. This is equivalent to 11x3+5x3, not 11x3-5x3.

Choice C is incorrect. This is equivalent to 11x6-5x6, not 11x3-5x3

Choice D is incorrect. This is equivalent to 11x6+5x6, not 11x3-5x3

Question 91 91 of 479 selected Equivalent Expressions E

Which expression is equivalent to 50x2+5x2?

  1. 250 x 2

  2. 10 x 2

  3. 45 x 2

  4. 55 x 2

Show Answer Correct Answer: D

Choice D is correct. The given expression shows addition of two like terms. Therefore, the given expression is equivalent to (50+5)x2, or 55x2.

Choice A is incorrect. This expression is equivalent to (50)(5)x2, not (50+5)x2.

Choice B is incorrect. This expression is equivalent to (505)x2, not (50+5)x2.

Choice C is incorrect. This expression is equivalent to (50-5)x2, not (50+5)x2.

Question 92 92 of 479 selected Nonlinear Functions E
x y
0 0
1 1
2 8
3 27

The table shown includes some values of x and their corresponding values of y. Which of the following graphs in the xy-plane could represent the relationship between x and y ?

  1.  

    The answer choice presents the graph of a function in the x y plane. The origin is labeled O, and the numbers negative 4 and 4 are indicated on each axis. The graph of the function is a curve. It begins above the x axis and to the left of the y axis, moves downward and to the right, and extends below the x axis and to the right of the y axis. From left to right,  the curve moves downward, at first quickly, and passes through the point with coordinates, negative 1 comma 1. It then moves more slowly as it passes through the origin. After crossing the origin, the curve continues to move slowly downward, then more quickly, and passes through the point with coordinates, 1 comma negative 1. It then extends downward and to the right.

     

  2.  

    The answer choice presents the graph of a function in the x y plane. The origin is labeled O, and the numbers negative 4 and 4 are indicated on each axis. The graph of the function is a curve. It begins below the x axis and to the left of the y axis, moves upward and to the right, and extends above the x axis and to the right of the y axis. From left to right, the curve moves upward, at first quickly, passes through the point with coordinates, negative 1 comma negative 1, and then moves more slowly as it passes through the origin. After crossing the origin, the curve continues to move slowly upward, then more quickly as it passes through the point with coordinates, 1 comma 1. It then extends upward and to the right.

     

  3.  

    The answer choice presents the graph of a function in the x y plane. The origin is labeled O, and the numbers negative 4 and 4 are indicated on each axis. The graph of the function is a curve. It begins below the x axis and to the left of the y axis, moves upward and to the right, and extends above the x axis and to the right of the y axis. From left to right, the curve moves upward, at first quickly, crosses the x axis at negative 1, and then moves more slowly as it crosses the y axis at 1. After crossing the y axis, the curve continues to move slowly upward, then moves more quickly as it passes through the point with coordinates, 1 comma 2. It then extends upward and to the right.

     

  4.  

    The answer choice presents the graph of a function in the x y plane. The origin is labeled O, and the numbers negative 4 and 4 are indicated on each axis. The graph of the function is a curve. It begins below the x axis and to the left of the y axis, moves upward and to the right, and extends above the x axis and to the right of the y axis. From left to right, the curve moves upward, at first quickly, passes through the point with coordinates, negative 1 comma negative 3, and then moves more slowly as it passes through the origin. After crossing the origin, the curve continues to move slowly upward, then moves more quickly as it passes through the point with coordinates, 1 comma 3. It then extends upward and to the right.

     

Show Answer Correct Answer: B

Choice B is correct. Each pair of values shown in the table gives the ordered pair of coordinates for a point that lies on the graph that represents the relationship between x and y in the xy-plane: the point with coordinates 0 comma 0, the point with coordinates 1 comma 1, the point with coordinates 2 comma 8, and the point with coordinates 3 comma 27. Only the graph in choice B passes through the points listed in the table that are visible in the given choices.

Choices A, C, and D are incorrect. None of these graphs passes through the point with coordinates 1 comma 1.

 

Question 93 93 of 479 selected Equivalent Expressions H

The expression (3x-23)(19x+6) is equivalent to the expression ax2+bx+c, where a , b , and c are constants. What is the value of b ?

Show Answer Correct Answer: -419

The correct answer is -419 . It's given that the expression (3x-23)(19x+6) is equivalent to the expression a x 2 + b x + c , where a , b , and c are constants. Applying the distributive property to the given expression, (3x-23)(19x+6), yields (3x)(19x)+(3x)(6)-(23)(19x)-(23)(6), which can be rewritten as 57x2+18x-437x-138. Combining like terms yields 57x2-419x-138. Since this expression is equivalent to a x 2 + b x + c , it follows that the value of b is -419 .

Question 94 94 of 479 selected Nonlinear Functions H

  • The parabola opens upward.
  • The vertex is at point (negative 1 comma negative 8).
  • The parabola passes through the following points:
    • (negative 2 comma negative 6)
    • (negative 1 comma negative 8) 
    • (0 comma negative 6)

The graph of y = 2 x 2 + b x + c is shown, where b and c are constants. What is the value of b c ?

Show Answer Correct Answer: -24

The correct answer is -24 . Since the graph passes through the point (0,-6), it follows that when the value of x is 0 , the value of y is -6 . Substituting 0 for x and -6 for y in the given equation yields -6=2(0)2+b(0)+c, or -6=c. Therefore, the value of c is -6 . Substituting -6 for c in the given equation yields y=2x2+bx-6. Since the graph passes through the point (-1,-8), it follows that when the value of x is -1 , the value of y is -8 . Substituting -1 for x and -8 for y in the equation y=2x2+bx-6 yields -8=2(-1)2+b(-1)-6, or -8=2-b-6, which is equivalent to -8=-4-b. Adding 4 to each side of this equation yields -4=-b. Dividing each side of this equation by -1 yields 4=b. Since the value of b is 4 and the value of c is -6 , it follows that the value of bc is (4)(-6), or -24 .

Alternate approach: The given equation represents a parabola in the xy-plane with a vertex at (-1,-8). Therefore, the given equation, y=2x2+bx+c, which is written in standard form, can be written in vertex form, y=a(x-h)2+k, where (h,k) is the vertex of the parabola and a is the value of the coefficient on the x2 term when the equation is written in standard form. It follows that a=2. Substituting 2 for a , -1 for h , and -8 for k in this equation yields y=2(x-(-1))2+(-8), or y=2(x+1)2-8. Squaring the binomial on the right-hand side of this equation yields y=2(x2+2x+1)-8. Multiplying each term inside the parentheses on the right-hand side of this equation by 2 yields y=2x2+4x+2-8, which is equivalent to y=2x2+4x-6. From the given equation y=2x2+bx+c, it follows that the value of b is 4 and the value of c is -6 . Therefore, the value of bc is (4)(-6), or -24 .

Question 95 95 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

( x - 47 ) 2 = 1

What is the sum of the solutions to the given equation?

Show Answer Correct Answer: 94

The correct answer is 94. Taking the square root of each side of the given equation yields x-47=1 or x-47=-1. Adding 47 to both sides of the equation x-47=1 yields x=48. Adding 47 to both sides of the equation x-47=-1 yields x=46. Therefore, the sum of the solutions to the given equation is 48+46, or 94.

Question 96 96 of 479 selected Nonlinear Functions M

  • The parabola opens downward.
  • The vertex is at point (4 comma 9).
  • The parabola passes through the following points:
    • (0 comma 5)
    • (2 comma 8)
    • (4 comma 9)
    • (6 comma 8)

The graph models the number of active projects a company was working on x months after the end of November 2012 , where 0x6. According to the model, what is the predicted number of active projects the company was working on at the end of November 2012 ?

  1. 0

  2. 5

  3. 8

  4. 9

Show Answer Correct Answer: B

Choice B is correct. It's given that the graph models the number of active projects a company was working on x months after the end of November 2012. Therefore, the value of x that corresponds to the end of November 2012 is 0 . The point at which x = 0 is the y-intercept of the graph. It follows that the y-intercept of the graph shown is the point (0,5). Therefore, according to the model, the predicted number of active projects the company was working on at the end of November 2012 is 5 .

Choice A is incorrect. This is the value of x that corresponds to the end of November 2012, not the predicted number of active projects the company was working on at the end of November 2012.

Choice C is incorrect. This is the predicted number of active projects the company was working on 2 months after the end of November 2012.

Choice D is incorrect. This is the predicted number of active projects the company was working on 4 months after the end of November 2012.

Question 97 97 of 479 selected Equivalent Expressions E

Which expression is equivalent to 20w-(4w+3w)?

  1. 10 w

  2. 13 w

  3. 19 w

  4. 21 w

Show Answer Correct Answer: B

Choice B is correct. Combining like terms inside the parentheses of the given expression, 20w-(4w+3w), yields 20w-(7w). Combining like terms in this resulting expression yields 13 w .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 98 98 of 479 selected Equivalent Expressions H

Which expression is equivalent to 4 4 x - 5 - 1 x + 1 ?

  1. 1 ( x + 1 ) ( 4 x - 5 )

  2. 3 3 x - 6

  3. - 1 ( x + 1 ) ( 4 x - 5 )

  4. 9 ( x + 1 ) ( 4 x - 5 )

Show Answer Correct Answer: D

Choice D is correct. The expression 44x-5-1x+1 can be rewritten as 44x-5+(-1)x+1. To add the two terms of this expression, the terms can be rewritten with a common denominator. Since x+1x+1=1, the expression 44x-5 can be rewritten as (x+1)(4)(x+1)(4x-5). Since 4x-54x-5=1, the expression -1x+1 can be rewritten as (4x-5)(-1)(4x-5)(x+1). Therefore, the expression 44x-5+(-1)x+1 can be rewritten as (x+1)(4)(x+1)(4x-5)+(4x-5)(-1)(4x-5)(x+1), which is equivalent to (x+1)(4)+(4x-5)(-1)(x+1)(4x-5). Applying the distributive property to each term of the numerator yields (4x+4)+(-4x+5)(x+1)(4x-5), or  (4x+(-4x))+(4+5)(x+1)(4x-5). Adding like terms in the numerator yields 9(x+1)(4x-5).

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 99 99 of 479 selected Nonlinear Functions H

The function f is defined by f(x)= ( x - 6 ) ( x - 2 ) ( x + 6 ) . In the xy-plane, the graph of y=g(x) is the result of translating the graph of y=f(x) up 4 units. What is the value of g(0)?

Show Answer Correct Answer: 76

The correct answer is 76 . It's given that the graph of y=g(x) is the result of translating the graph of y=f(x) up 4 units in the xy-plane. It follows that the graph of y=g(x) is the same as the graph of y=f(x)+4. Substituting g(x) for y in the equation y=f(x)+4 yields g(x)=f(x)+4. It's given that f(x)=(x-6)(x-2)(x+6). Substituting (x-6)(x-2)(x+6) for f(x) in the equation g(x)=f(x)+4 yields g(x)=(x-6)(x-2)(x+6)+4. Substituting 0 for x in this equation yields g(0)=(0-6)(0-2)(0+6)+4, or g(0)=76. Thus, the value of g(0) is 76 .

Question 100 100 of 479 selected Equivalent Expressions E

Which of the following is equivalent to 3 times, open parenthesis, x plus 5, close parenthesis, minus 6 ?

  1. 3 x minus 3

  2. 3 x minus 1

  3. 3 x plus 9

  4. 15 x minus 6

Show Answer Correct Answer: C

Choice C is correct. Using the distributive property to multiply 3 and open parenthesis, x plus 5, close parenthesis gives 3 x plus 15, minus 6, which can be rewritten as 3 x plus 9.

Choice A is incorrect and may result from rewriting the given expression as 3 times, open parenthesis, x plus 5, minus 6, close parenthesis. Choice B is incorrect and may result from incorrectly rewriting the expression as open parenthesis, 3 x plus 5, close parenthesis, minus 6. Choice D is incorrect and may result from incorrectly rewriting the expression as 3 times 5 x, minus 6.

 

Question 101 101 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

  • For the curve in the system:
    • Moving from left to right:
      • The curve passes from quadrant 2 to quadrant 1.
      • In quadrant 2, the curve trends up gradually to point (0 comma 3.25).
      • In quadrant 1, the curve trends up sharply.
    • The curve passes through the following points:
      • (1 comma 3.5)
      • (2 comma 4)
      • (4 comma 7)
  • For the line in the system:
    • The line slants gradually up from left to right.
    • The line passes through the following points:
      • (2 comma 4)
      • (7 comma 7.5)

The graph of a system of a linear equation and a nonlinear equation is shown. What is the solution (x,y) to this system?

  1. (0,0)

  2. (0,2)

  3. (2,4)

  4. (4,0)

Show Answer Correct Answer: C

Choice C is correct. The solution to the system of two equations corresponds to the point where the graphs of the equations intersect. The graphs of the linear equation and the nonlinear equation shown intersect at the point (2,4). Thus, the solution to the system is (2,4).

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 102 102 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

open parenthesis, x minus 4, close parenthesis, times, open parenthesis, x plus 2, close parenthesis, times, open parenthesis, x minus 1, close parenthesis, equals 0

What is the product of the solutions to the given equation?

  1. 8

  2. 3

  3. negative 3

  4. negative 8

Show Answer Correct Answer: D

Choice D is correct. By the zero-product property, if open parenthesis, x minus 4, close parenthesis, times, open parenthesis, x plus 2, close parenthesis, times, open parenthesis, x minus 1, close parenthesis, equals 0, then x minus 4, equals 0, x plus 2, equals 0, or x minus 1, equals 0. Solving each of these equations for x yields x equals 4, x equals negative 2, or x equals 1. The product of these solutions is 4 times negative 2, times 1, equals negative 8.

Choice A is incorrect and may result from sign errors made when solving the given equation. Choice B is incorrect and may result from finding the sum, not the product, of the solutions. Choice C is incorrect and may result from finding the sum, not the product, of the solutions in addition to making sign errors when solving the given equation.

 

Question 103 103 of 479 selected Equivalent Expressions H

the fraction with numerator x squared, minus c, and denominator x minus b, end fraction

In the expression above, b and c are positive integers. If the expression is equivalent to x plus b and x is not equal to b, which of the following could be the value of c ?

  1. 4

  2. 6

  3. 8

  4. 10

Show Answer Correct Answer: A

Choice A is correct. If the given expression is equivalent to x plus b, then the fraction with numerator x squared, minus c, and denominator x minus b, end fraction, equals, x plus b, where x isn’t equal to b. Multiplying both sides of this equation by x  minus b yields x squared, minus c, equals, open parenthesis, x plus b, close parenthesis, times, open parenthesis, x minus b, close parenthesis. Since the right-hand side of this equation is in factored form for the difference of squares, the value of c must be a perfect square. Only choice A gives a perfect square for the value of c.

Choices B, C, and D are incorrect. None of these values of c produces a difference of squares. For example, when 6 is substituted for c in the given expression, the result is the fraction with numerator x squared, minus 6, and denominator x minus b, end fraction. The expression x squared, minus 6 can’t be factored with integer values, and therefore the fraction with numerator x squared, minus 6, and denominator x minus b, end fraction isn’t equivalent to x plus b.

 

Question 104 104 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

the fraction with numerator 2 times, open parenthesis, x plus 1, close parenthesis, and denominator x plus 5, end fraction, equals 1 minus the fraction with numerator 1, and denominator x plus 5, end fraction

What is the solution to the equation above?

  1. 0

  2. 2

  3. 3

  4. 5

Show Answer Correct Answer: B

Choice B is correct. Since the fraction with numerator x plus 5, and denominator x plus 5, end fraction is equivalent to 1, the right-hand side of the given equation can be rewritten as the fraction with numerator x plus 5, and denominator x plus 5, end fraction, minus, the fraction 1 over, x plus 5, end fraction , or the fraction with numerator x plus 4, and denominator x plus 5, end fraction. Since the left- and right-hand sides of the equation the fraction with numerator 2 times, open parenthesis, x plus 1, close parenthesis, and denominator x plus 5, end fraction, equals, the fraction with numerator x plus 4, and denominator x plus 5, end fraction have the same denominator, it follows that 2 times, open parenthesis, x plus 1, close parenthesis, equals x plus 4. Applying the distributive property of multiplication to the expression 2 times, open parenthesis, x plus 1, close parenthesis yields 2 times x, plus, 2 times 1, or 2 x plus 2. Therefore, 2 x plus 2, equals, x plus 4. Subtracting x and 2 from both sides of this equation yields x equals 2.

Choices A, C, and D are incorrect. If x equals 0, then the fraction with numerator 2 times, open parenthesis, 0 plus 1, close parenthesis, and denominator 0 plus 5, end fraction, equals 1 minus, the fraction with numerator 1, and denominator 0 plus 5, end fraction. This can be rewritten as 2 over 5, equals 4 over 5, which is a false statement. Therefore, 0 isn’t a solution to the given equation. Substituting 3 and 5 into the given equation yields similarly false statements.

 

Question 105 105 of 479 selected Equivalent Expressions M

g(x)=35x+76

h(x)=6x-5

The functions g and h are defined by the equations shown. Which expression is equivalent to g(x)·h(x)?

  1. 18 x 2 5 - 35 6

  2. 18 x 2 5 + 27 x 11 - 35 6

  3. 18 x 2 5 - 4 x - 35 6

  4. 18 x 2 5 + 4 x - 35 6

Show Answer Correct Answer: D

Choice D is correct. It’s given that g(x)=35x+76 and h(x)=6x-5. Substituting 35x+76 for g(x) and 6 x - 5 for h(x) in the expression g(x)·h(x) yields (35x+76)(6x-5). This expression can be rewritten as 35x(6x-5)+76(6x-5), or 18x25-3x+7x-356, which is equivalent to 18x25+4x-356.

Choice A is incorrect. This expression is equivalent to 35x(6x)+76(-5), not (35x+76)(6x-5).

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This expression is equivalent to (35x-76)(6x+5), not (35x+76)(6x-5).

Question 106 106 of 479 selected Nonlinear Functions M

A rectangle has a length of x units and a width of (x-15) units. If the rectangle has an area of 76 square units, what is the value of x ?

  1. 4

  2. 19

  3. 23

  4. 76

Show Answer Correct Answer: B

Choice B is correct. The area of a rectangle is equal to its length multiplied by its width. Multiplying the given length, x units, by the given width, (x-15) units, yields x(x-15) square units. If the rectangle has an area of 76 square units, it follows that x(x-15)=76, or x2-15x=76. Subtracting 76 from both sides of this equation yields x2-15x-76=0. Factoring the left-hand side of this equation yields (x-19)(x+4)=0. Applying the zero product property to this equation yields two solutions: x = 19 and x = -4 . Since x is the rectangle’s length, in units, which must be positive, the value of x is 19 .

Choice A is incorrect. This is the width, in units, of the rectangle, not the value of x .

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the area, in square units, of the rectangle, not the value of x .

Question 107 107 of 479 selected Nonlinear Functions M

A scientist initially measures 12,000 bacteria in a growth medium. 4 hours later, the scientist measures 24,000 bacteria. Assuming exponential growth, the formula P=C(2)rt gives the number of bacteria in the growth medium, where r and C are constants and P is the number of bacteria t  hours after the initial measurement. What is the value of r ?

  1. 1 12,000

  2. 1 4

  3. 4

  4. 12,000

Show Answer Correct Answer: B

Choice B is correct. It’s given that the formula P=C(2)rt gives the number of bacteria in a growth medium, where r and C are constants and P is the number of bacteria t hours after the initial measurement. It’s also given that a scientist initially measures 12,000 bacteria in the growth medium. Since the initial measurement is 0 hours after the initial measurement, it follows that when t = 0 , P = 12,000 . Substituting 0 for t and 12,000 for P in the given equation yields 12,000=C(2)r(0), or 12,000=C(2)0, which is equivalent to 12,000=C.  It’s given that 4 hours later, the scientist measures 24,000 bacteria, or when t = 4 , P = 24,000 . Substituting 4 for t , 24,000 for P , and 12,000 for C in the given equation yields 24,000=12,000(2)4r. Dividing each side of this equation by 12,000 yields 2 = 2 4 r , or 21=24r, which is equivalent to 1 = 4 r . Dividing both sides of this equation by 4 yields 1 4 = r . Therefore, the value of r is 1 4 .

Choice A is incorrect. This is the value of the reciprocal of C .

Choice C is incorrect. This is the value of the reciprocal of r .

Choice D is incorrect. This is the value of C .

Question 108 108 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

( x - 2 ) 2 = 3 x + 34

What is the smallest solution to the given equation?

Show Answer Correct Answer: -3

The correct answer is -3. Squaring both sides of the given equation yields (x-2)2=3x+34, which can be rewritten as x2-4x+4=3x+34. Subtracting 3 x and 34 from both sides of this equation yields x2-7x-30=0. This quadratic equation can be rewritten as (x-10)(x+3)=0. According to the zero product property, (x-10)(x+3) equals zero when either x-10=0 or x+3=0. Solving each of these equations for x yields x=10 or x=-3. Therefore, the given equation has two solutions, 10 and -3 . Of these two solutions, -3 is the smallest solution to the given equation.

Question 109 109 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

y equals, x squared, minus 4 x, plus 4, and y equals, 4 minus x

If the ordered pair x comma y satisfies the system of equations above, what is one possible value of x ?

Show Answer

The correct answer is either 0 or 3. For an ordered pair to satisfy a system of equations, both the x- and y-values of the ordered pair must satisfy each equation in the system. Both expressions on the right-hand side of the given equations are equal to y, therefore it follows that both expressions on the right-hand side of the equations are equal to each other: x squared, minus 4 x, plus 4, equals, 4 minus x. This equation can be rewritten as x squared, minus 3 x, equals 0, and then through factoring, the equation becomes x times, open parenthesis, x minus 3, close parenthesis, equals 0. Because the product of the two factors is equal to 0, it can be concluded that either x equals 0 or x minus 3, equals 0, or rather, x equals 0 or x equals 3. Note that 0 and 3 are examples of ways to enter a correct answer.

Question 110 110 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

x 2 + y + 10 = 10

8x+16-y=0

The solution to the given system of equations is (x,y). What is the value of x ?

  1. -16

  2. -4

  3. 2

  4. 8

Show Answer Correct Answer: B

Choice B is correct. Adding y to each side of the second equation in the given system of equations yields 8 x + 16 = y . Substituting 8 x + 16 for y in the first equation yields x2+8x+16+10=10. Subtracting 10 from each side of this equation yields x 2 + 8 x + 16 = 0 . This equation can be rewritten as (x+4)2=0. Taking the square root of each side of this equation yields x + 4 = 0 . Subtracting 4 from each side of this equation yields x = - 4 . Therefore, the value of x is - 4 .

Choice A is incorrect. This is the value of y , not x .

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 111 111 of 479 selected Nonlinear Functions H

A quadratic function models a projectile's height, in meters, above the ground in terms of the time, in seconds, after it was launched. The model estimates that the projectile was launched from an initial height of 7 meters above the ground and reached a maximum height of 51.1 meters above the ground 3 seconds after the launch. How many seconds after the launch does the model estimate that the projectile will return to a height of 7 meters?

  1. 3

  2. 6

  3. 7

  4. 9

Show Answer Correct Answer: B

Choice B is correct. It's given that a quadratic function models the projectile's height, in meters, above the ground in terms of the time, in seconds, after it was launched. It follows that an equation representing the model can be written in the form f(x)=a(x-h)2+k, where f(x) is the projectile's estimated height above the ground, in meters, x seconds after the launch, a is a constant, and k is the maximum height above the ground, in meters, the model estimates the projectile reached h seconds after the launch. It's given that the model estimates the projectile reached a maximum height of 51.1 meters above the ground 3 seconds after the launch. Therefore, k = 51.1 and h = 3 . Substituting 51.1 for k and 3 for h in the equation f(x)=a(x-h)2+k yields f(x)=a(x-3)2+51.1. It's also given that the model estimates that the projectile was launched from an initial height of 7 meters above the ground. Therefore, when x = 0 f(x)=7. Substituting 0 for x and 7 for f(x) in the equation f(x)=a(x-3)2+51.1 yields 7=a(0-3)2+51.1, or 7=9a+51.1. Subtracting 51.1 from both sides of this equation yields -44.1 = 9 a . Dividing both sides of this equation by 9 yields -4.9 = a . Substituting -4.9 for a in the equation f(x)=a(x-3)2+51.1 yields f(x)=-4.9(x-3)2+51.1. Therefore, the equation f(x)=-4.9(x-3)2+51.1 models the projectile's height, in meters, above the ground x seconds after it was launched. The number of seconds after the launch that the model estimates that the projectile will return to a height of 7 meters is the value of x when f(x)=7. Substituting 7 for f(x) in f(x)=-4.9(x-3)2+51.1 yields 7=-4.9(x-3)2+51.1. Subtracting 51.1 from both sides of this equation yields -44.1=-4.9(x-3)2. Dividing both sides of this equation by -4.9 yields 9=(x-3)2. Taking the square root of both sides of this equation yields two equations: 3 = x - 3 and -3 = x - 3 . Adding 3 to both sides of the equation 3 = x - 3 yields 6 = x . Adding 3 to both sides of the equation -3 = x - 3 yields 0 = x . Since 0 seconds after the launch represents the time at which the projectile was launched, 6 must be the number of seconds the model estimates that the projectile will return to a height of 7 meters.

Alternate approach: It's given that a quadratic function models the projectile's height, in meters, above the ground in terms of the time, in seconds, after it was launched. It's also given that the model estimates that the projectile was launched from an initial height of 7 meters above the ground and reached a maximum height of 51.1 meters above the ground 3 seconds after the launch. Since the model is quadratic, and quadratic functions are symmetric, the model estimates that for any given height less than the maximum height, the time the projectile takes to travel from the given height to the maximum height is the same as the time the projectile takes to travel from the maximum height back to the given height. Thus, since the model estimates the projectile took 3 seconds to travel from 7 meters above the ground to its maximum height of 51.1 meters above the ground, the model also estimates the projectile will take 3 more seconds to travel from its maximum height of 51.1 meters above the ground back to 7 meters above the ground. Thus, the model estimates that the projectile will return to a height of 7 meters 3 seconds after it reaches its maximum height, which is 6 seconds after the launch.

Choice A is incorrect. The model estimates that 3 seconds after the launch, the projectile reached a height of 51.1 meters, not 7 meters.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 112 112 of 479 selected Nonlinear Functions H

y = x 2 - 14 x + 22

The given equation relates the variables x and y . For what value of x does the value of y reach its minimum?

Show Answer Correct Answer: 7

The correct answer is 7 . When an equation is of the form y=ax2+bx+c, where a , b , and c are constants, the value of y reaches its minimum when x=-b2a. Since the given equation is of the form y=ax2+bx+c, it follows that a = 1 , b = -14 , and c = 22 . Therefore, the value of y reaches its minimum when x=-(-14)2(1), or x = 7 .

Question 113 113 of 479 selected Equivalent Expressions M

the cube root of x to the power 3, times y to the power 6, end root

Which of the following expressions is equivalent to the expression above?

  1. y squared
  2. x times y squared
  3. y cubed
  4. x times y cubed
Show Answer Correct Answer: B

Choice B is correct. One of the properties of radicals is the nth root of a times b, end root, equals the nth root of a, end root, times the nth root of b, end root. Thus, the given expression can be rewritten as the cube root of x cubed, end root, times the cube root of y to the power 6, end root. Simplifying by taking the cube root of each part gives x1 ⋅ y2, or xy2.

Choices A, C, and D are incorrect and may be the result of incorrect application of the properties of exponents and radicals.

Question 114 114 of 479 selected Equivalent Expressions M

Which expression is equivalent to (d-6)(8d2-3)?

  1. 8 d 3 - 14 d 2 - 3 d + 18

  2. 8 d 3 - 17 d 2 + 48

  3. 8 d 3 - 48 d 2 - 3 d + 18

  4. 8 d 3 - 51 d 2 + 48

Show Answer Correct Answer: C

Choice C is correct. Applying the distributive property to the given expression yields d(8d2-3)-6(8d2-3). Applying the distributive property once again to this expression yields (d)(8d2)+(d)(-3)+(-6)(8d2)+(-6)(-3), or 8d3+(-3d)+(-48d2)+18. This expression can be rewritten as 8d3-48d2-3d+18. Thus, (d-6)(8d2-3) is equivalent to 8d3-48d2-3d+18.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 115 115 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

An oceanographer uses the equation s equals, three halves p to model the speed s, in knots, of an ocean wave, where p represents the period of the wave, in seconds. Which of the following represents the period of the wave in terms of the speed of the wave?

  1. p equals, two thirds s

  2. p equals, three halves s

  3. p equals, two thirds plus s

  4. p equals, three halves plus s

Show Answer Correct Answer: A

Choice A is correct. It’s given that p represents the period of the ocean wave, so the equation s equals, three halves p can be solved for p to represent the period of the wave in terms of the speed of the wave. Multiplying both sides of the equation by the reciprocal of three halves will isolate p. This yields two thirds s equals, two thirds times, three halves p, which simplifies to two thirds s, equals p. Therefore, p equals, two thirds s.

Choices B, C, and D are incorrect and may result from errors made when rearranging the equation to solve for p.

 

Question 116 116 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

If 42x=16, what is the value of 6 x ?

  1. 24

  2. 48

  3. 72

  4. 96

Show Answer Correct Answer: B

Choice B is correct. Dividing each side of the given equation by 4 yields 2x=4. Squaring both sides of this equation yields 2 x = 16 . Multiplying each side of this equation by 3 yields 6 x = 48 . Therefore, the value of 6 x is 48 .

Choice A is incorrect. This is the value of 3 x , not 6 x .

Choice C is incorrect. This is the value of 9 x , not 6 x .

Choice D is incorrect. This is the value of 12 x , not 6 x .

Question 117 117 of 479 selected Nonlinear Functions E

The function f is defined by f(x)=16x. What is the value of f(x) when x = 3 ?

  1. 1 3

  2. 1 6

  3. 1 9

  4. 1 18

Show Answer Correct Answer: D

Choice D is correct. It's given that f(x)=16x. Substituting 3 for x in this equation yields f(3)=16(3), or f(3)=118. Therefore, when x = 3 , the value of f(x) is 118.

Choice A is incorrect. This is the value of f(x) when x=0.5.

Choice B is incorrect. This is the value of f(x) when x = 1

Choice C is incorrect. This is the value of f(x) when x=1.5.

Question 118 118 of 479 selected Equivalent Expressions M

If x squared equals, a, plus b and y squared equals, a, plus c, which of the following is equal to open parenthesis, x squared, minus y squared, close parenthesis, squared ?

  1. a, squared, minus 2 a, c, plus c squared

  2. b squared, minus 2 b c, plus c squared

  3. 4, a, squared, minus 4 a, b c, plus c squared

  4. 4 a, squared, minus 2 a, b c, plus b squared c squared

Show Answer Correct Answer: B

Choice B is correct. It’s given that x squared equals, a, plus b and y squared equals, a, plus c. Using the distributive property, the expression open parenthesis, x squared, minus y squared, close parenthesis, squared can be rewritten as open parenthesis, x squared, close parenthesis, squared, minus 2 x squared, times y squared, plus, open parenthesis, y squared, close parenthesis, squared. Substituting a, plus b and a, plus c for x squared and y squared, respectively, in this expression yields open parenthesis, a, plus b, close parenthesis, squared, minus, 2 times, open parenthesis, open parenthesis, a, plus b, close parenthesis, times, open parenthesis, a, plus c, close parenthesis, close parenthesis, plus, open parenthesis, a, plus c, close parenthesis, squared. Expanding this expression yields open parenthesis, a, squared, plus 2 a, b, plus b squared, close parenthesis, minus, open parenthesis, 2 a, squared, plus 2 b c, plus 2 a, c, plus 2 a, b, close parenthesis, plus, open parenthesis, a, squared, plus 2 a, c, plus c squared, close parenthesis. Combining like terms, this expression can be rewritten as b squared, minus 2 b c, plus c squared.

Choices A, C, and D are incorrect and may result from an error in using the distributive property, substituting, or combining like terms.

 

Question 119 119 of 479 selected Nonlinear Functions M

f(x)=x5+9x+17

For the given function f , the graph of y=f(x) in the xy-plane passes through the point (0,b), where b is a constant. What is the value of b ?

Show Answer Correct Answer: 17

The correct answer is 17. It's given that the graph of y=f(x) in the xy-plane passes through the point (0,b), where b is a constant. It follows that f(0) equals b. Substituting 0 for x in the given function yields f(0)=05+9(0)+17, or f(0)=17. Therefore, the value of b is 17.

Question 120 120 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

Which of the following is a solution to the equation 2 x squared, minus 4, equals x squared ?

  1. 1

  2. 2

  3. 3

  4. 4

Show Answer Correct Answer: B

Choice B is correct. Subtracting x2 from both sides of the given equation yields x2 – 4 = 0. Adding 4 to both sides of the equation gives x2 = 4. Taking the square root of both sides of the equation gives x = 2 or x = –2. Therefore, x = 2 is one solution to the original equation.

Alternative approach: Subtracting x2 from both sides of the given equation yields x2 – 4 = 0. Factoring this equation gives x2 – 4 = (x + 2)(x2) = 0, such that x = 2 or x = 2. Therefore, x = 2 is one solution to the original equation.

Choices A, C, and D are incorrect and may be the result of computation errors.

Question 121 121 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

q-29r=s

The given equation relates the positive numbers q , r , and s . Which equation correctly expresses q in terms of r and s ?

  1. q=s-29r

  2. q=s+29r

  3. q=29rs

  4. q=-s29r

Show Answer Correct Answer: B

Choice B is correct. Adding 29 r to each side of the given equation yields q=s+29r. Therefore, the equation q=s+29r correctly expresses q in terms of r and s .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 122 122 of 479 selected Nonlinear Functions H

A park ranger hung squirrel houses each in the shape of a right rectangular prism for fox squirrels. Each house has a height of 11 inches. The length of each house's base is x inches, which is 1 inch more than the width of the house's base. Which function V gives the volume of each house, in cubic inches, in terms of the length of the house's base?

  1. V(x)=11x(x-1)

  2. V(x)=11x(x+1)

  3. V(x)=x(x+11)(x-1)

  4. V(x)=x(x+11)(x+1)

Show Answer Correct Answer: A

Choice A is correct. The volume of a prism is equal to the area of its base times its height. It's given that the length of each house's base is x inches and that this length is 1 inch more than the width, in inches, of the house's base. It follows that the width, in inches, of the house's base is x-1. The area of a rectangle is the product of its length and its width. Therefore, the area of the base of the house is x(x-1) square inches. It's given that the height of each house is 11 inches. Therefore, the function V that gives the volume of each house, in cubic inches, in terms of the length of the house's base x is V(x)=x(x-1)11, or V(x)=11x(x-1).

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 123 123 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

57x2+(57b+a)x+ab=0

In the given equation, a and b are positive constants. The product of the solutions to the given equation is kab, where k is a constant. What is the value of k ?

  1. 1 57

  2. 1 19

  3. 1

  4. 57

Show Answer Correct Answer: A

Choice A is correct. The left-hand side of the given equation is the expression 57x2+(57b+a)x+ab. Applying the distributive property to this expression yields 57x2+57bx+ax+ab. Since the first two terms of this expression have a common factor of 57 x and the last two terms of this expression have a common factor of a , this expression can be rewritten as 57x(x+b)+a(x+b). Since the two terms of this expression have a common factor of (x+b), it can be rewritten as (x+b)(57x+a). Therefore, the given equation can be rewritten as (x+b)(57x+a)=0. By the zero product property, it follows that x+b=0 or 57x+a=0. Subtracting b from both sides of the equation x+b=0 yields x=-b. Subtracting a from both sides of the equation 57x+a=0 yields 57x=-a. Dividing both sides of this equation by 57 yields x=-a57. Therefore, the solutions to the given equation are - b and -a57. It follows that the product of the solutions of the given equation is (-b)(-a57), or ab57. It’s given that the product of the solutions of the given equation is kab. It follows that ab57=kab, which can also be written as ab(157)=ab(k). It’s given that a and b are positive constants. Therefore, dividing both sides of the equation ab(157)=ab(k) by a b yields 157=k. Thus, the value of k is 157.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 124 124 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

- x 2 + b x - 676 = 0

In the given equation, b is a positive integer. The equation has no real solution. What is the greatest possible value of b ?

Show Answer Correct Answer: 51

The correct answer is 51 . A quadratic equation of the form a x 2 + b x + c = 0 , where a , b , and c are constants, has no real solution if and only if its discriminant, - 4 a c + b 2 , is negative. In the given equation, a = -1 and c = -676 . Substituting -1 for a and -676 for c in this expression yields a discriminant of b2-4(-1)(-676), or b 2 - 2,704 . Since this value must be negative, b2-2,704<0, or b2<2,704. Taking the positive square root of each side of this inequality yields b<52. Since b is a positive integer, and the greatest integer less than 52 is 51 , the greatest possible value of b is 51 .

Question 125 125 of 479 selected Nonlinear Functions H

Function f is defined by f(x)=-ax+b, where a and b are constants. In the xy-plane, the graph of y=f(x)-15 has a y-intercept at (0,-997). The product of a and b is 65 7 . What is the value of a ?

Show Answer Correct Answer: 5

The correct answer is 5 . It’s given that f(x)=-ax+b. Substituting -ax+b for f(x) in the equation y=f(x)-15 yields y=-ax+b-15. It’s given that the y-intercept of the graph of y=f(x)-15 is (0,-997). Substituting 0 for x and -997 for y in the equation y=-ax+b-15 yields -997=-a0+b-15, which is equivalent to -997=-1+b-15, or -997=b-16. Adding 16 to both sides of this equation yields 137=b. It’s given that the product of a and b is 65 7 , or a b = 65 7 . Substituting 13 7   for b in this equation yields (a)(137)=657. Dividing both sides of this equation by 13 7 yields a = 5 .

Question 126 126 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

p = k 4 j + 9

The given equation relates the distinct positive numbers p , k , and j . Which equation correctly expresses 4 j + 9 in terms of p and k ?

  1. 4 j + 9 = k p

  2. 4 j + 9 = k p

  3. 4j+9=k-p

  4. 4 j + 9 = p k

Show Answer Correct Answer: A

Choice A is correct. To express 4j+9 in terms of p and k , the given equation must be solved for 4j+9. Since it's given that j is a positive number, 4j+9 is not equal to zero. Therefore, multiplying both sides of the given equation by 4j+9 yields the equivalent equation p(4j+9)=k. Since it's given that p is a positive number, p is not equal to zero. Therefore, dividing each side of the equation p(4j+9)=k by p yields the equivalent equation 4j+9=kp.

Choice B is incorrect. This equation is equivalent to p=4j+9k.

Choice C is incorrect. This equation is equivalent to p=k-4j-9.

Choice D is incorrect. This equation is equivalent to p=k(4j+9).

Question 127 127 of 479 selected Equivalent Expressions M

If x is not equal to 0, which of the following expressions is equivalent to the fraction with numerator the square root of 16 x to the fourth power, y to the eighth power, end root, and denominator x cubed, end fraction ?

  1. 8 x squared, y to the fourth power

  2. 4 x, y to the fourth power

  3. 4 x raised to the negative 2 power, y squared

  4. 4 x raised to the negative 1 power, y to the fourth power

Show Answer Correct Answer: D

Choice D is correct. Taking the square root of an exponential expression halves the exponent, so the fraction with numerator the square root of, 16 x to the fourth power times y to the eighth power, end root, and denominator x cubed, end fraction, equals, the fraction with numerator 4 x squared, times y to the fourth power, and denominator x cubed, which further reduces to the fraction with numerator 4, y to the fourth power, and denominator x. This can be rewritten as 4, x to the negative 1 power, times y to the fourth power.

Choice A is incorrect and may result from neglecting the denominator of the given expression and from incorrectly calculating the square root of 16. Choice B is incorrect and may result from rewriting 1 over x as x to the first power rather than x to the negative 1 power. Choice C is incorrect and may result from taking the square root of the variables in the numerator twice instead of once.

 

Question 128 128 of 479 selected Nonlinear Functions H

The function g is defined by g(x)=(x+14)(t-x), where t is a constant. In the xy-plane, the graph of y=g(x) passes through the point (24,0). What is the value of g(0)?

Show Answer Correct Answer: 336

The correct answer is 336 . By the zero product property, if (x+14)(t-x)=0, then x+14=0, which gives x = -14 , or (t-x)=0, which gives x = t . Therefore, g(x)=0 when x = -14 and when x = t . Since the graph of y=g(x) passes through the point (24,0), it follows that g(24)=0, so t = 24 . Substituting 24 for t in the equation g(x)=(x+14)(t-x) yields g(x)=(x+14)(24-x). The value of g(0) can be calculated by substituting 0 for x in this equation, which yields g(0)=(0+14)(24-0), or g(0)=336.

Question 129 129 of 479 selected Nonlinear Functions M

g(x)=x2+55

What is the minimum value of the given function?

  1. 0

  2. 55

  3. 110

  4. 3,025

Show Answer Correct Answer: B

Choice B is correct. For a quadratic function defined by an equation of the form g(x)=a(x-h)2+k, where a , h , and k are constants and a>0, the minimum value of the function is k . In the given function, a = 1 , h = 0 , and k = 55 . Therefore, the minimum value of the given function is 55 .

Choice A is incorrect. This is the value of x for which the given function reaches its minimum value, not the minimum value of the function.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 130 130 of 479 selected Nonlinear Functions H

The area of a rectangular banner is 2,661 square inches. The banner's length x , in inches, is 24 inches longer than its width, in inches. Which equation represents this situation?

  1. 0 = x 2 - 24 x - 2,661

  2. 0 = x 2 - 24 x + 2,661

  3. 0 = x 2 + 24 x - 2,661

  4. 0 = x 2 + 24 x + 2,661

Show Answer Correct Answer: A

Choice A is correct. It’s given that the banner’s length x , in inches, is 24 inches longer than its width, in inches. It follows that the banner’s width, in inches, can be represented by the expression x - 24 . The area of a rectangle is the product of its length and its width. It's given that the area of the banner is 2,661 square inches, so it follows that 2,661 = x ( x - 24 ) , or 2,661 = x 2 - 24 x . Subtracting 2,661 from each side of this equation yields 0 = x 2 - 24 x - 2,661 . Therefore, the equation that represents this situation is 0 = x 2 - 24 x - 2,661 .

Choice B is incorrect and may result from representing the width, in inches, of the banner as 24-x, rather than x - 24 .

Choice C is incorrect and may result from representing the width, in inches, of the banner as x + 24 , rather than x - 24 .

Choice D is incorrect and may result from conceptual or calculation errors.

Question 131 131 of 479 selected Nonlinear Functions H

f(x)= ( x - 14 ) ( x + 19 )

The function f is defined by the given equation. For what value of x does f(x) reach its minimum?

  1. -266

  2. -19

  3. - 33 2

  4. - 5 2

Show Answer Correct Answer: D

Choice D is correct. It's given that f(x)=(x-14)(x+19), which can be rewritten as f(x)=x2+5x-266. Since the coefficient of the x2-term is positive, the graph of y=f(x) in the xy-plane opens upward and reaches its minimum value at its vertex. The x-coordinate of the vertex is the value of x such that f(x) reaches its minimum. For an equation in the form f(x)=ax2+bx+c, where a , b , and c are constants, the x-coordinate of the vertex is -b2a. For the equation f(x)=x2+5x-266, a=1, b=5, and c=-266. It follows that the x-coordinate of the vertex is -52(1), or -52. Therefore, f(x) reaches its minimum when the value of x is -52.

Alternate approach: The value of x for the vertex of a parabola is the x-value of the midpoint between the two x-intercepts of the parabola. Since it’s given that f(x)=(x-14)(x+19), it follows that the two x-intercepts of the graph of y=f(x) in the xy-plane occur when x=14 and x=-19, or at the points (14,0) and (-19,0). The midpoint between two points, (x1,y1) and (x2,y2), is (x1+x22,y1+y22). Therefore, the midpoint between (14,0) and (-19,0) is (14+(-19)2,0+02), or (-52,0). It follows that f(x) reaches its minimum when the value of x is -52.

Choice A is incorrect. This is the y-coordinate of the y-intercept of the graph of y=f(x) in the xy-plane.

Choice B is incorrect. This is one of the x-coordinates of the x-intercepts of the graph of y=f(x) in the xy-plane.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 132 132 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

If 3 x 2 - 18 x - 15 = 0 , what is the value of x 2 - 6 x ?

Show Answer Correct Answer: 5

The correct answer is 5 . Dividing each side of the given equation by 3  yields x2-6x-5=0. Adding 5 to each side of this equation yields x2-6x=5. Therefore, if 3x2-18x-15=0, the value of x2-6x is 5 .

Question 133 133 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

In the xy-plane, what is the y-coordinate of the point of intersection of the graphs of y equals, open parenthesis, x minus 1, close parenthesis, squared and y equals, 2 x minus 3 ?

Show Answer

The correct answer is 1. The point of intersection of the graphs of the given equations is the solution to the system of the two equations. Since y equals, open parenthesis, x minus 1, close parenthesis, squared and y equals, 2 x minus 3, it follows that open parenthesis, x minus 1, close parenthesis, squared, equals, 2 x minus 3, or open parenthesis, x minus 1, close parenthesis, times, open parenthesis, x minus 1, close parenthesis, equals, 2 x minus 3. Applying the distributive property to the left-hand side of this equation yields x squared, minus 2 x, plus 1, equals, 2 x minus 3. Subtracting 2 x from and adding 3 to both sides of this equation yields x squared, minus 4 x, plus 4, equals 0. Factoring the left-hand side of this equation yields open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x minus 2, close parenthesis, equals 0. By the zero product property, if open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x minus 2, close parenthesis, equals 0, it follows that x minus 2, equals 0. Adding 2 to both sides of x minus 2, equals 0 yields x equals 2. Substituting 2 for x in either of the given equations yields y equals 1. For example, substituting 2 for x in the second given equation yields y equals, 2 times 2, minus 3, or y equals 1. Therefore, the point of intersection of the graphs of the given equations is the point with coordinates 2 comma 1. The y-coordinate of this point is 1.

Question 134 134 of 479 selected Nonlinear Functions M

A sample of a certain isotope takes 29 years to decay to half its original mass. The function s(t)=184(0.5)t29 gives the approximate mass of this isotope, in grams, that remains t years after a 184 -gram sample starts to decay. Which statement is the best interpretation of s(87)=23 in this context?

  1. Approximately 23 grams of the sample remains 87 years after the sample starts to decay.

  2. The mass of the sample has decreased by approximately 23 grams 87 years after the sample starts to decay.

  3. The mass of the sample has decreased by approximately 87 grams 23 years after the sample starts to decay.

  4. Approximately 87 grams of the sample remains 23 years after the sample starts to decay.

Show Answer Correct Answer: A

Choice A is correct. In the given function, s(t) represents the approximate mass, in grams, of the sample that remains t years after the sample starts to decay. It follows that the best interpretation of s(87)=23 is that approximately 23 grams of the sample remains 87 years after the sample starts to decay.

Choice B is incorrect. The mass of the sample has decreased by approximately 184-23, or 161 , grams, not 23 grams, 87 years after the sample starts to decay.

Choice C is incorrect. The mass of the sample has decreased by approximately 78 grams, not 87 grams, 23 years after the sample starts to decay.

Choice D is incorrect. This would be the best interpretation of s(23)=87, not s(87)=23.

Question 135 135 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

y+k=x+26

y-k=x2-5x

In the given system of equations, k is a constant. The system has exactly one distinct real solution. What is the value of k ?

Show Answer Correct Answer: 17.5, 35/2

The correct answer is 352. Subtracting the second equation from the first equation yields (y+k)-(y-k)=x+26-(x2-5x), or 2k=-x2+6x+26. This is equivalent to x2-6x+(2k-26)=0. It's given that the system has exactly one distinct real solution; therefore, this equation has exactly one distinct real solution. An equation of the form ax2+bx+c=0, where a , b , and c are constants, has exactly one distinct real solution when the discriminant, b2-4ac, is equal to 0 . The equation x2-6x+(2k-26)=0 is of this form, where a=1, b=-6, and c=2k-26. Substituting these values into the discriminant, b2-4ac, yields (-6)2-4(1)(2k-26). Setting the discriminant equal to 0 yields (-6)2-4(1)(2k-26)=0, or -8k+140=0. Subtracting 140 from both sides of this equation yields -8k=-140. Dividing both sides of this equation by - 8 yields k=352. Note that 35/2 and 17.5 are examples of ways to enter a correct answer.

Question 136 136 of 479 selected Nonlinear Functions H

The functions f and g are defined by the given equations.

f(x)=3+|-2x-x2|

g(w)=|-ww-1|-w+5

If f(-4)=c, where c is a constant, what is the value of g(c)?

Show Answer Correct Answer: -4.9, -49/10

The correct answer is - 4.9 . The value of f(-4) is the value of f(x) when x=-4. Substituting - 4 for x in the equation f(x)=3+|-2x-x2| yields f(-4)=3+|-2(-4)-(-4)2|, or f(-4)=3+|-8|, which is equivalent to f(-4)=3+8, or f(-4)=11. Since it's given that f(-4)=c, it follows that c = 11 and the value of g(c) is the value of g(11). Substituting 11 for w in the equation g(w)=|-ww-1|-w+5 yields g(11)=|-1111-1|-11+5, or g(11)=|-1.1|-6, which is equivalent to g(11)=1.1-6, or g(11)=-4.9. Note that -4.9 and -49/10 are examples of ways to enter a correct answer.

Question 137 137 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

Which quadratic equation has no real solutions?

  1. x 2 + 14 x - 49 = 0

  2. x 2 - 14 x + 49 = 0

  3. 5 x 2 - 14 x - 49 = 0

  4. 5 x 2 - 14 x + 49 = 0

Show Answer Correct Answer: D

Choice D is correct. The number of solutions to a quadratic equation in the form ax2+bx+c=0, where a , b , and c are constants, can be determined by the value of the discriminant, b2-4ac. If the value of the discriminant is greater than zero, then the quadratic equation has two distinct real solutions. If the value of the discriminant is equal to zero, then the quadratic equation has exactly one real solution. If the value of the discriminant is less than zero, then the quadratic equation has no real solutions. For the quadratic equation in choice D, 5x2-14x+49=0, a = 5 , b = -14 , and c = 49 . Substituting 5 for a , -14 for b , and 49 for c in b2-4ac yields (-14)2-4(5)(49), or -784 . Since -784 is less than zero, it follows that the quadratic equation 5x2-14x+49=0 has no real solutions.

Choice A is incorrect. The value of the discriminant for this quadratic equation is 392 . Since 392 is greater than zero, it follows that this quadratic equation has two real solutions.

Choice B is incorrect. The value of the discriminant for this quadratic equation is 0 . Since zero is equal to zero, it follows that this quadratic equation has exactly one real solution.

Choice C is incorrect. The value of the discriminant for this quadratic equation is 1,176. Since 1,176 is greater than zero, it follows that this quadratic equation has two real solutions.

Question 138 138 of 479 selected Nonlinear Functions E

  • Moving from left to right:
    • The curve passes from quadrant 2 to quadrant 1.
    • In quadrant 2, the curve trends up gradually to point (0 comma 2).
    • In quadrant 1, the curve trends up sharply.
  • The curve passes through the following points:
    • (negative 1 comma 1)
    • (0 comma 2)
    • (2 comma 8)

What is the y-intercept of the graph shown?

  1. (0,0)

  2. (0,2)

  3. (2,0)

  4. (2,2)

Show Answer Correct Answer: B

Choice B is correct. The y-intercept of a graph in the xy-plane is the point at which the graph crosses the y-axis. The graph shown crosses the y-axis at the point (0,2). Therefore, the y-intercept of the graph shown is (0,2)

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 139 139 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

5 | x | = 45

What is the positive solution to the given equation?

Show Answer Correct Answer: 9

The correct answer is 9. Dividing both sides of the given equation by 5 yields |x|=9. By the definition of absolute value, if |x|=9, then x=9 or x=-9. Therefore, the two solutions to the equation 5|x|=45 are 9 and -9. It follows that the positive solution to the given equation is 9.

Question 140 140 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

2 x squared, minus 4 x, equals t

In the equation above, t is a constant. If the equation has no real solutions, which of the following could be the value of t ?

  1. negative 3

  2. negative 1

  3. 1

  4. 3

Show Answer Correct Answer: A

Choice A is correct. The number of solutions to any quadratic equation in the form a, x squared, plus b x, plus c, equals 0, where a, b, and c are constants, can be found by evaluating the expression b squared, minus 4 a, c, which is called the discriminant. If the value of b squared, minus 4 a, c is a positive number, then there will be exactly two real solutions to the equation. If the value of b squared, minus 4 a, c is zero, then there will be exactly one real solution to the equation. Finally, if the value of b squared, minus 4 a, c is negative, then there will be no real solutions to the equation.

The given equation 2 x squared, minus 4 x, equals t is a quadratic equation in one variable, where t is a constant. Subtracting t from both sides of the equation gives 2 x squared, minus 4 x, minus t, equals 0. In this form, a, equals 2, b equals negative 4, and c equals negative t. The values of t for which the equation has no real solutions are the same values of t for which the discriminant of this equation is a negative value. The discriminant is equal to open parenthesis, negative 4, close parenthesis, squared, minus 4 times 2, times negative t; therefore, open parenthesis, negative 4, close parenthesis, squared, minus, 4 times 2, times negative t, is less than 0. Simplifying the left side of the inequality gives 16 plus 8, t, is less than 0. Subtracting 16 from both sides of the inequality and then dividing both sides by 8 gives t is less than negative 2. Of the values given in the options, negative 3 is the only value that is less than negative 2. Therefore, choice A must be the correct answer.

Choices B, C, and D are incorrect and may result from a misconception about how to use the discriminant to determine the number of solutions of a quadratic equation in one variable.

 

Question 141 141 of 479 selected Nonlinear Functions E

P(t)=1,800(1.02)t

The function P gives the estimated number of marine mammals in a certain area, where t is the number of years since a study began. What is the best interpretation of P(0)=1,800 in this context?

  1. The estimated number of marine mammals in the area was 102 when the study began.

  2. The estimated number of marine mammals in the area was 1,800 when the study began.

  3. The estimated number of marine mammals in the area increased by 102 each year during the study.

  4. The estimated number of marine mammals in the area increased by 1,800 each year during the study.

Show Answer Correct Answer: B

Choice B is correct. The function P gives the estimated number of marine mammals in a certain area, where t is the number of years since a study began. Since the value of P(0) is the value of P(t) when t = 0 , it follows that P(0)=1,800 means that the value of P(t) is 1,800 when t = 0 . Since t is the number of years since the study began, it follows that t = 0 is 0 years since the study began, or when the study began. Therefore, the best interpretation of P(0)=1,800 in this context is the estimated number of marine mammals in the area was 1,800 when the study began.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 142 142 of 479 selected Nonlinear Functions E

  • The parabola opens upward.
  • The vertex is at point (4 comma 0).
  • The parabola passes through the following points:
    • (3 comma 5)
    • (4 comma 0)
    • (5 comma 5)

What is the x-intercept of the graph shown?

  1. (-5,0)

  2. (5,0)

  3. (-4,0)

  4. (4,0)

Show Answer Correct Answer: D

Choice D is correct. The x-intercept of the graph shown is the point (x,y) on the graph where y = 0 . At y = 0 , the corresponding value of x is 4 . Therefore, the x-intercept of the graph shown is (4,0).

Choice A is incorrect. This is the x-intercept of a graph in the xy-plane that intersects the x-axis at x = -5 , not x = 4 .

Choice B is incorrect. This is the x-intercept of a graph in the xy-plane that intersects the x-axis at x = 5 , not x = 4 .

Choice C is incorrect. This is the x-intercept of a graph in the xy-plane that intersects the x-axis at x = -4 , not x = 4 .

Question 143 143 of 479 selected Nonlinear Functions M

q(x)=32(2x)

Which table gives three values of x and their corresponding values of q(x) for function q ?

Show Answer Correct Answer: D

Choice D is correct. Substituting -1 for x in the given function yields q(-1)=32(2)-1, which is equivalent to q(-1)=32(12), or q(-1)=16. Therefore, when x = -1 , the corresponding value of q(x) for function q is 16 . Substituting 0 for x in the given function yields q(0)=32(2)0, which is equivalent to q(0)=32(1), or q(0)=32. Therefore, when x = 0 , the corresponding value of q(x) for function q is 32 . Substituting 1 for x in the given function yields q(1)=32(2)1, which is equivalent to q(1)=32(2), or q(1)=64. Therefore, when x = 1 , the corresponding value of q(x) for function q is 64 . Of the choices given, only the table in choice D gives these three values of x and their corresponding values of q(x) for function q .

Choice A is incorrect. This table gives three values of x and their corresponding values of q(x) for the function q(x)=32(2x).

Choice B is incorrect. This table gives three values of x and their corresponding values of q(x) for the function q(x)=2(32)x.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 144 144 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

14 j + 5 k = m

The given equation relates the numbers j , k , and m . Which equation correctly expresses k in terms of j and m ?

  1. k=m-14j5

  2. k=15m-14j

  3. k=14j-m5

  4. k=5m-14j

Show Answer Correct Answer: A

Choice A is correct. Subtracting 14j from each side of the given equation results in 5k=m-14j. Dividing each side of this equation by 5 results in k=m-14j5.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 145 145 of 479 selected Nonlinear Functions M

f(x)=(x+6)(x-4)

If the given function f is graphed in the xy-plane, where y=f(x), what is the x-coordinate of an x-intercept of the graph?

Show Answer Correct Answer: -6, 4

The correct answer is either -6 or 4 . The x-intercepts of a graph in the xy-plane are the points where y = 0 . Thus, for an x-intercept of the graph of y=f(x), 0=f(x). Substituting 0 for f(x) in the equation f(x)=(x+6)(x-4) yields 0=(x+6)(x-4). By the zero product property, x + 6 = 0 and x - 4 = 0 . Subtracting 6 from both sides of the equation x + 6 = 0 yields x = -6 . Adding 4 to both sides of the equation x - 4 = 0 yields x = 4 . Therefore, the x-coordinates of the x-intercepts of the graph of y=f(x) are -6 and 4 . Note that -6 and 4 are examples of ways to enter a correct answer.

Question 146 146 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

The solutions to x 2 + 6 x + 7 = 0 are r and s , where r<s. The solutions to x 2 + 8 x + 8 = 0 are t and u , where t<u. The solutions to x 2 + 14 x + c = 0 , where c is a constant, are r+t and s+u. What is the value of c ?

Show Answer Correct Answer: 31

The correct answer is 31 . Subtracting 7 from both sides of the equation x 2 + 6 x + 7 = 0 yields x 2 + 6 x = - 7 . To complete the square, adding (62)2, or 32, to both sides of this equation yields x2+6x+32=-7+32, or ( x + 3 ) 2 = 2 . Taking the square root of both sides of this equation yields x+3=±2. Subtracting 3 from both sides of this equation yields x=-3±2. Therefore, the solutions r and s to the equation x 2 + 6 x + 7 = 0 are -3-2 and -3+2. Since r<s, it follows that r=-3-2 and s=-3+2. Subtracting 8 from both sides of the equation x 2 + 8 x + 8 = 0 yields x 2 + 8 x = - 8 . To complete the square, adding (82)2, or 42, to both sides of this equation yields x2+8x+42=-8+42, or ( x + 4 ) 2 = 8 . Taking the square root of both sides of this equation yields x+4=±8, or x+4=±22. Subtracting 4 from both sides of this equation yields x=-4±22. Therefore, the solutions t and u to the equation x 2 + 8 x + 8 = 0 are -4-22 and -4+22. Since t<u, it follows that t=-4-22 and u=-4+22. It's given that the solutions to x 2 + 14 x + c = 0 , where c is a constant, are r + t and s + u . It follows that this equation can be written as (x-(r+t))(x-(s+u))=0, which is equivalent to x2-(r+t+s+u)x+(r+t)(s+u)=0. Therefore, the value of c is (r+t)(s+u). Substituting -3-2 for r -4-22 for t -3+2 for s , and -4+22 for u in this equation yields ((-3-2)+(-4-22))((-3+2)+(-4+22)), which is equivalent to (-7-32)(-7+32), or (-7)(-7)-(32)(32), which is equivalent to 49-18, or 31 . Therefore, the value of c is 31 .

Question 147 147 of 479 selected Equivalent Expressions E

Which expression is equivalent to 23 x 3 + 2 x 2 + 9 x ?

  1. 23x(x2+2x+9)

  2. 9x(23x3+2x2+1)

  3. x(23x2+2x+9)

  4. 34(x3+x2+x)

Show Answer Correct Answer: C

Choice C is correct. Since x is a common factor of each term in the given expression, the given expression can be rewritten as x(23x2+2x+9).

Choice A is incorrect. This expression is equivalent to 23x3+46x2+207x.

Choice B is incorrect. This expression is equivalent to 207x4+18x3+9x.

Choice D is incorrect. This expression is equivalent to 34x3+34x2+34x.

Question 148 148 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

x ( k x - 56 ) = -16

In the given equation, k is an integer constant. If the equation has no real solution, what is the least possible value of k ?

Show Answer Correct Answer: 50

The correct answer is 50 . An equation of the form ax2+bx+c=0, where a , b , and c are constants, has no real solutions if and only if its discriminant, b2-4ac, is negative. Applying the distributive property to the left-hand side of the equation x(kx-56)=-16 yields kx2-56x=-16. Adding 16 to each side of this equation yields kx2-56x+16=0. Substituting k for a , -56 for b , and 16 for c in b2-4ac yields a discriminant of (-56)2-4(k)(16), or 3,136-64k. If the given equation has no real solution, it follows that the value of 3,136-64k must be negative. Therefore, 3,136-64k<0. Adding 64 k to both sides of this inequality yields 3,136<64k. Dividing both sides of this inequality by 64 yields 49<k, or k>49. Since it's given that k is an integer, the least possible value of k is 50

Question 149 149 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

5(x+7)=15(x-17)(x+7)

What is the sum of the solutions to the given equation?

Show Answer Correct Answer: 10.33, 31/3

The correct answer is 31 3 . Subtracting 5(x+7) from each side of the given equation yields 0=15(x-17)(x+7)-5(x+7). Since 5(x+7) is a common factor of each of the terms on the right-hand side of this equation, it can be rewritten as 0=5(x+7)(3(x-17)-1). This is equivalent to 0=5(x+7)(3x-51-1), or 0=5(x+7)(3x-52). Dividing both sides of this equation by 5 yields 0=(x+7)(3x-52). Since a product of two factors is equal to 0 if and only if at least one of the factors is 0 , either x + 7 = 0 or 3 x - 52 = 0 . Subtracting 7 from both sides of the equation x + 7 = 0 yields x = - 7 . Adding 52 to both sides of the equation 3 x - 52 = 0 yields 3 x = 52 . Dividing both sides of this equation by 3 yields x = 52 3 . Therefore, the solutions to the given equation are - 7 and 52 3 . It follows that the sum of the solutions to the given equation is -7+523, which is equivalent to -213+523, or 31 3 . Note that 31/3 and 10.33 are examples of ways to enter a correct answer.

Question 150 150 of 479 selected Nonlinear Functions E

At the time of posting a video, a social media channel had 53 subscribers. Each day for five days after the video was posted, the number of subscribers doubled from the number the previous day. Which equation gives the total number of subscribers, n , to the channel d days after the video was posted?

  1. n=(53)d

  2. n=53(2)d

  3. n=53(12)d

  4. n=(53)2+d

Show Answer Correct Answer: B

Choice B is correct. It's given that each day for five days after a social media channel posted a video, the number of subscribers doubled from the number the previous day. Since the number of subscribers doubled each day, the relationship between n and d can be represented by an exponential equation of the form n=abd, where a is the number of subscribers at the time of posting the video and the number of subscribers to the channel increases by a factor of b each day. It's given that at the time of posting the video, the channel had 53 subscribers. Therefore, a=53. It's also given that the number of subscribers doubled, or increased by a factor of 2, from the number the previous day. Therefore, b=2. Substituting 53 for a and 2 for b in the equation n=abd yields n=53(2)d.

Choice A is incorrect. This equation gives the total number of subscribers to a channel for which the initial number of subscribers was 1 and the number increased each day by 53 times the number from the previous day.

Choice C is incorrect. This equation gives the total number of subscribers to a channel for which the number of subscribers each day was half the number from the previous day, rather than double the number.

Choice D is incorrect and may result from conceptual errors.

Question 151 151 of 479 selected Nonlinear Functions M

The function f is defined by f(x)=4x-1. What is the value of f(21)?

  1. -84

  2. 1 84

  3. 4 21

  4. 21 4

Show Answer Correct Answer: C

Choice C is correct. It’s given that function f is defined by the equation f(x)=4x-1. The value of f(21) is the value of f(x) when x=21. Substituting 21 for x in the given equation yields f(21)=4(21)-1, which is equivalent to f(21)=4(121), or f(21)=421.

Choice A is incorrect. This is the value of f(21) when f(x)=-4x, rather than f(x)=4x-1.

Choice B is incorrect. This is the value of f(21) when f(x)=(4x)-1, rather than f(x)=4x-1.

Choice D is incorrect. This is the value of f(21) when f(x)=(4-1)x, rather than f(x)=4x-1.

Question 152 152 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

(x+2)(x-5)(x+9)=0

What is a positive solution to the given equation?

  1. 3

  2. 4

  3. 5

  4. 18

Show Answer Correct Answer: C

Choice C is correct. Applying the zero product property to the given equation yields three equations: x + 2 = 0 , x - 5 = 0 , and x + 9 = 0 . Subtracting 2 from both sides of the equation x + 2 = 0 yields x = -2 . Adding 5 to both sides of the equation x - 5 = 0 yields x = 5 . Subtracting 9 from both sides of the equation x + 9 = 0 yields x = -9 . Therefore, the solutions to the given equation are -2 , 5 , and -9 . It follows that a positive solution to the given equation is 5 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

 

Question 153 153 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

- 2 x 2 + 20 x + c = 0

In the given equation, c is a constant. The equation has exactly one solution. What is the value of c ?

  1. -68

  2. -50

  3. -32

  4. 0

Show Answer Correct Answer: B

Choice B is correct. It's given that the equation -2x2+20x+c=0, where c is a constant, has exactly one solution. A quadratic equation of the form ax2+bx+c=0 has exactly one solution if and only if its discriminant, b2-4ac, is equal to zero. It follows that for the given equation, a=-2 and b=20. Substituting -2 for a and 20 for b in b2-4ac yields 202-4(-2)(c), or 400+8c. Since the discriminant must equal zero, it follows that 400+8c=0. Subtracting 400 from both sides of this equation yields 8c=-400. Dividing each side of this equation by 8 yields c=-50. Therefore, the value of c is -50.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 154 154 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

- 4 x 2 - 7 x = -36

What is the positive solution to the given equation?

  1. 7 4

  2. 9 4

  3. 4

  4. 7

Show Answer Correct Answer: B

Choice B is correct. Multiplying each side of the given equation by -16 yields 64 x 2 + 112 x = 576 . To complete the square, adding 49 to each side of this equation yields 64x2+112x+49=576+49, or (8x+7)2=625. Taking the square root of each side of this equation yields two equations: 8 x + 7 = 25 and 8 x + 7 = -25 . Subtracting 7 from each side of the equation 8 x + 7 = 25 yields 8 x = 18 . Dividing each side of this equation by 8 yields x=188, or x = 9 4 . Therefore, 9 4 is a solution to the given equation. Subtracting 7 from each side of the equation 8 x + 7 = -25 yields 8 x = -32 . Dividing each side of this equation by 8 yields x = -4 . Therefore, the given equation has two solutions, 9 4 and -4 . Since 9 4 is positive, it follows that 9 4 is the positive solution to the given equation.

Alternate approach: Adding 4 x 2 and 7 x to each side of the given equation yields 0 = 4 x 2 + 7 x - 36 . The right-hand side of this equation can be rewritten as 4x2+16x-9x-36. Factoring out the common factor of 4 x from the first two terms of this expression and the common factor of -9 from the second two terms yields 4x(x+4)-9(x+4). Factoring out the common factor of (x+4) from these two terms yields the expression (4x-9)(x+4). Since this expression is equal to 0 , it follows that either 4 x - 9 = 0 or x + 4 = 0 . Adding 9 to each side of the equation 4 x - 9 = 0 yields 4 x = 9 . Dividing each side of this equation by 4 yields x = 9 4 . Therefore, 9 4 is a positive solution to the given equation. Subtracting 4 from each side of the equation x + 4 = 0 yields x = -4 . Therefore, the given equation has two solutions, 9 4 and -4 . Since 9 4 is positive, it follows that 9 4 is the positive solution to the given equation.

Choice A is incorrect. Substituting 7 4 for x in the given equation yields -492=-36, which is false.

Choice C is incorrect. Substituting 4 for x in the given equation yields -92=-36, which is false.

Choice D is incorrect. Substituting 7 for x in the given equation yields -245=-36, which is false.

Question 155 155 of 479 selected Equivalent Expressions E

Which expression is equivalent to 19(x2-7)?

  1. 19 x 2 - 133

  2. 19 x 2 - 26

  3. 19 x 2 - 7

  4. 19 x 2 + 12

Show Answer Correct Answer: A

Choice A is correct. The expression 19(x2-7) can be rewritten as 19(x2)-19(7), which is equivalent to 19 x 2 - 133 .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 156 156 of 479 selected Nonlinear Functions M

Immanuel purchased a certain rare coin on January 1. The function f(x)=65(1.03)x, where 0x10, gives the predicted value, in dollars, of the rare coin x years after Immanuel purchased it. What is the best interpretation of the statement “f(8) is approximately equal to 82 ” in this context?

  1. When the rare coin's predicted value is approximately 82 dollars, it is 8 % greater than the predicted value, in dollars, on January 1 of the previous year.

  2. When the rare coin’s predicted value is approximately 82 dollars, it is 8 times the predicted value, in dollars, on January 1 of the previous year.

  3. From the day Immanuel purchased the rare coin to 8 years after Immanuel purchased the coin, its predicted value increased by a total of approximately 82 dollars.

  4. 8 years after Immanuel purchased the rare coin, its predicted value is approximately 82 dollars.

Show Answer Correct Answer: D

Choice D is correct. It’s given that the function f(x)=65(1.03)x gives the predicted value, in dollars, of a certain rare coin x years after Immanuel purchased it. It follows that f(x) represents the predicted value, in dollars, of the coin x years after Immanuel purchased it. Since the value of f(8) is the value of f(x) when x = 8 , it follows that “f(8) is approximately equal to 82 ” means that f(x) is approximately equal to 82 when x = 8 . Therefore, the best interpretation of the statement “f(8) is approximately equal to 82 ” in this context is 8 years after Immanuel purchased the rare coin, its predicted value is approximately 82 dollars.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Question 157 157 of 479 selected Nonlinear Functions H

The function f gives the product of a number, x , and a number that is 91 more than x . Which equation defines f ?

  1. f(x)=x2+x+91

  2. f(x)=x2+91

  3. f(x)=x2+91x

  4. f(x)=x2+91x+91

Show Answer Correct Answer: C

Choice C is correct. It’s given that the function f gives the product of a number, x, and a number that is 91 more than x. A number that is 91 more than x can be represented by the expression x+91. Therefore, f can be defined by the equation f(x)=x(x+91), or f(x)=x2+91x.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 158 158 of 479 selected Nonlinear Functions H

A model estimates that at the end of each year from 2015 to 2020, the number of squirrels in a population was 150 % more than the number of squirrels in the population at the end of the previous year. The model estimates that at the end of 2016, there were 180 squirrels in the population. Which of the following equations represents this model, where n is the estimated number of squirrels in the population t years after the end of 2015 and t5?

  1. n=72(1.5)t

  2. n=72(2.5)t

  3. n=180(1.5)t

  4. n=180(2.5)t

Show Answer Correct Answer: B

Choice B is correct. Since the model estimates that the number of squirrels in the population increased by a fixed percentage, 150%, each year, the model can be represented by an exponential equation of the form n=a(1+p100)t, where a is the estimated number of squirrels in the population at the end of 2015, and the model estimates that at the end of each year, the number is p% more than the number at the end of the previous year. Since the model estimates that at the end of each year, the number was 150% more than the number at the end of the previous year, p = 150 . Substituting 150 for p in the equation n=a(1+p100)t yields n=a(1+150100)t, which is equivalent to n=a(1+1.5)t, or n=a(2.5)t. It’s given that the estimated number of squirrels at the end of 2016 was 180 . This means that when t = 1 , n = 180 . Substituting 1 for t and 180 for n in the equation n=a(2.5)t yields 180=a(2.5)1, or 180 = 2.5 a . Dividing each side of this equation by 2.5 yields 72 = a . Substituting 72 for a in the equation n=a(2.5)t yields n=72(2.5)t.

Choice A is incorrect. This equation represents a model where at the end of each year, the estimated number of squirrels was 150% of, not 150% more than, the estimated number at the end of the previous year.

Choice C is incorrect. This equation represents a model where at the end of each year, the estimated number of squirrels was 150% of, not 150% more than, the estimated number at the end of the previous year, and the estimated number of squirrels at the end of 2015, not the end of 2016, was 180 .

Choice D is incorrect. This equation represents a model where the estimated number of squirrels at the end of 2015, not the end of 2016, was 180 .

Question 159 159 of 479 selected Equivalent Expressions H

0.36 x 2 + 0.63 x + 1.17

The given expression can be rewritten as a(4x2+7x+13), where a is a constant. What is the value of a ?

Show Answer Correct Answer: .09, 9/100

The correct answer is .09. It's given that the expression 0.36x2+0.63x+1.17 can be rewritten as a(4x2+7x+13). Applying the distributive property to the expression a(4x2+7x+13) yields 4ax2+7ax+13a. Therefore, 0.36x2+0.63x+1.17 can be rewritten as 4ax2+7ax+13a. It follows that in the expressions 0.36x2+0.63x+1.17 and 4ax2+7ax+13a, the coefficients of x2 are equivalent, the coefficients of x are equivalent, and the constant terms are equivalent. Therefore, 0.36 = 4 a , 0.63 = 7 a , and 1.17 = 13 a . Solving any of these equations for a yields the value of a . Dividing both sides of the equation 0.36 = 4 a by 4 yields 0.09 = a . Therefore, the value of a is 0.09 . Note that .09 and 9/100 are examples of ways to enter a correct answer.

Question 160 160 of 479 selected Nonlinear Functions H

f(x)=|59-2x|

The function f is defined by the given equation. For which of the following values of k does f(k)=3k?

  1. 59 5

  2. 59 2

  3. 177 5

  4. 59

Show Answer Correct Answer: A

Choice A is correct. The value of k for which f(k)=3k can be found by substituting k for x and 3 k for f(x) in the given equation, f(x)=|59-2x|, which yields 3k=|59-2k|. For this equation to be true, either -3k=59-2k or 3k=59-2k. Adding 2 k to both sides of the equation -3k=59-2k yields -k=59. Dividing both sides of this equation by -1 yields k=-59. To check whether -59 is the value of k , substituting -59 for k in the equation 3k=|59-2k| yields 3(-59)=|59-2(-59)|, which is equivalent to -177=|177|, or -177=177, which isn't a true statement. Therefore, -59 isn't the value of k . Adding 2 k to both sides of the equation 3k=59-2k yields 5k=59. Dividing both sides of this equation by 5 yields k=595. To check whether 59 5 is the value of k , substituting 59 5  for k in the equation 3k=|59-2k| yields 3(595)=|59-2(595)|, which is equivalent to 1775=|1775|, or 1775=1775, which is a true statement. Therefore, the value of k for which f(k)=3k is 595.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 161 161 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

14x7y=2w+19

The given equation relates the distinct positive real numbers w , x , and y . Which equation correctly expresses w in terms of x and y ?

  1. w=xy-19

  2. w=28x14y-19

  3. w=(xy)2-19

  4. w=(28x14y)2-19

Show Answer Correct Answer: C

Choice C is correct. Dividing each side of the given equation by 2 yields 14x14y=2w+192, or xy=w+19. Because it's given that each of the variables is positive, squaring each side of this equation yields the equivalent equation (xy)2=w+19. Subtracting 19 from each side of this equation yields (xy)2-19=w, or w=(xy)2-19.

Choice A is incorrect. This equation isn't equivalent to the given equation. 

Choice B is incorrect. This equation isn't equivalent to the given equation. 

Choice D is incorrect. This equation isn't equivalent to the given equation. 

Question 162 162 of 479 selected Equivalent Expressions M

x squared, plus 6 x, plus 4

Which of the following is equivalent to the expression above?

  1. (x + 3)2 + 5

  2. (x + 3)2 – 5

  3. (x – 3)2 + 5

  4. (x – 3)2 – 5

Show Answer Correct Answer: B

Choice B is correct. The given quadratic expression is in standard form, and each answer choice is in vertex form. Completing the square converts the expression from standard form to vertex form. The first step is to rewrite the expression as follows: x squared, plus 6 x, plus 4, equals, x squared, plus 6 x, plus 9, plus 4, minus 9. The first three terms of the revised expression can be rewritten as a perfect square as follows: x squared, plus 6 x, plus 9, plus 4, minus 9, equals, open parenthesis, x plus 3, close parenthesis, squared, plus 4, minus 9. Combining the constant terms gives open parenthesis, x plus 3, close parenthesis, squared, minus 5.

Choice A is incorrect. Squaring the binomial and simplifying the expression in choice A gives x squared, plus 6 x, plus 9, plus 5. Combining like terms gives x squared, minus 6 x, plus 14, not x squared, plus 6 x, plus 4. Choice C is incorrect. Squaring the binomial and simplifying the expression in choice C gives x squared, minus 6 x, plus 9, plus 5. Combining like terms gives x squared, minus 6 x, plus 14, not x squared, plus 6 x, plus 4. Choice D is incorrect. Squaring the binomial and simplifying the expression in choice D gives x squared, minus 6 x, plus 9, minus 5. Combining like terms gives x squared, minus 6 x, plus 4, not x squared, plus 6 x, plus 4.

 

Question 163 163 of 479 selected Equivalent Expressions H

In the expression  , p is a constant. This expression is equivalent to the expression 6 x squared minus 155 x plus 24. What is the value of p ?

  1. negative 3

  2. 7

  3. 13

  4. 155

Show Answer Correct Answer: B

Choice B is correct. Using the distributive property, the first given expression can be rewritten as 6x2 + 3px + 24 – 16px – 64x + 24, and then rewritten as 6x2 + (3p – 16p – 64)x + 24. Since the expression 6x2 + (3p – 16p – 64)x + 24 is equivalent to 6x2 – 155x + 24, the coefficients of the x terms from each expression are equivalent to each other; thus 3p – 16p – 64 = 155. Combining like terms gives –13p  – 64 = 155. Adding 64 to both sides of the equation gives –13p = 71. Dividing both sides of the equation by –13 yields p = 7.

Choice A is incorrect. If p = 3, then the first expression would be equivalent to 6x2 – 25x + 24. Choice C is incorrect. If p = 13, then the first expression would be equivalent to 6x2 – 233x + 24. Choice D is incorrect. If p = 155, then the first expression would be equivalent to 6x2 – 2,079x + 24.

Question 164 164 of 479 selected Nonlinear Functions H

f(t)=55t-2t2

The function f is defined by the given equation. The function g is defined by g(t)=f(t)+3. Which expression represents the maximum value of g(t)?

  1. 3+(552)2

  2. 3+2(554)2

  3. 3-2(554)2

  4. 3-(552)2

Show Answer Correct Answer: B

Choice B is correct. It’s given that function g is defined by g(t)=f(t)+3 and that f(t)=55t-2t2. Substituting 55t-2t2 for f(t) in g(t)=f(t)+3 yields g(t)=55t-2t2+3, or g(t)=-2t2+55t+3. The maximum value of g(t) can be found by completing the square to rewrite the equation defining g in the form g(t)=a(t-h)2+k, where the maximum value of the function is k, which occurs when t=h, and a is a negative constant. The equation g(t)=-2t2+55t+3 is equivalent to g(t)=-2(t2-552t)+3, which can be rewritten as g(t)=-2(t2-552t+(554)2)+3+2(554)2, or g(t)=-2(t-554)2+3+2(554)2. This equation is in the form g(t)=a(t-h)2+k, where a=-2, h=554, and k=3+2(554)2. Thus, the maximum value of g(t) is 3+2(554)2.
Alternate approach: Since the function f is a quadratic function, the maximum value of f(t) occurs at the value of t that’s halfway between the two zeros of the function. The zeros of function f can be found by substituting 0 for f(t) in the equation defining f, which yields 0=55t-2t2. This equation can be rewritten as 0=t(55-2t). By the zero product property, it follows that t=0 or 55-2t=0. Subtracting 55 from each side of the equation 55-2t=0 yields -2t=-55. Dividing each side of this equation by -2 yields t=552. Therefore, the zeros of function f are 0 and 552. The value that’s halfway between 0 and 552 can be found by calculating the average of 0 and 552, which is 0+5522, or 554. It follows that the maximum of function f occurs when t=554. Substituting 554 for t in the equation defining function f yields f(554)=55(554)-2(554)2, which is equivalent to f(554)=5524-2(55242). Multiplying 5524 by 44 in this equation to get a common denominator yields f(554)=4(55242)-2(55242), or f(554)=2(55242), which is equivalent to f(554)=2(554)2. Thus, the maximum value of f(t) is 2(554)2. Since the equation defining g(t) is g(t)=f(t)+3, the maximum value of g(t) is 3 greater than the maximum value of f(t). It follows that the maximum value of g(t) is 3+2(554)2.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 165 165 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

open parenthesis, x plus 2, close parenthesis, times, open parenthesis, x plus 3, close parenthesis, equals, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x minus 3, close parenthesis, plus 10

Which of the following is a solution to the given equation?

  1. 1

  2. 0

  3. negative 2

  4. negative 5

Show Answer Correct Answer: A

Choice A is correct. Applying the distributive property on the left- and right-hand sides of the given equation yields x squared, plus 2 x, plus 3 x, plus 6, equals, x squared, minus 2 x, minus 3 x, plus 6, plus 10, or x squared, plus 5 x, plus 6, equals, x squared, minus 5 x, plus 16. Subtracting x squared from and adding 5 x to both sides of this equation yields 10 x plus 6, equals 16. Subtracting 6 from both sides of this equation and then dividing both sides by 10 yields x equals 1.

Choices B, C, and D are incorrect. Substituting 0, negative 2, or negative 5 for x in the given equation will result in a false statement. If x equals 0, the given equation becomes 6 equals 16; if x equals negative 2, the given equation becomes 0 equals 30; and if x equals negative 5, the given equation becomes 6 equals 66. Therefore, the values 0, negative 2, and negative 5 aren’t solutions to the given equation.

 

Question 166 166 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

y=x2+1.7

y=1.7-x

Which graph represents the given system of equations?

    • For the line in the system:
      • The line slants sharply down from left to right.
      • The line passes through the following points:
        • (negative 1 comma 2.7)
        • (0 comma 1.7)
    • For the parabola in the system:
      • The parabola opens upward.
      • The vertex is at point (0 comma 1.7).
      • The parabola passes through the following points:
        • (negative 1 comma 2.7)
        • (0 comma 1.7)
        • (1 comma 2.7)

    • For the line in the system:
      • The line slants sharply up from left to right.
      • The line passes through the following points:
        • (0 comma 1.7)
        • (1 comma 2.7)
    • For the parabola in the system:
      • The parabola opens upward.
      • The vertex is at point (0 comma 1.7).
      • The parabola passes through the following points:
        • (negative 1 comma 2.7)
        • (0 comma 1.7)
        • (1 comma 2.7)

    • For the line in the system:
      • The line slants sharply down from left to right.
      • The line passes through the following points:
        • (negative 1 comma 2.7)
        • (0 comma 1.7)
    • For the parabola in the system:
      • The parabola opens upward.
      • The vertex is at point (negative 1.7 comma 0).
      • The parabola passes through the following points:
        • (negative 2.7 comma 1)
        • (negative 1.7 comma 0)
        • (negative 0.7 comma 1)

    • For the line in the system:
      • The line slants sharply up from left to right.
      • The line passes through the following points:
        • (0 comma 1.7)
        • (1 comma 2.7)
    • For the parabola in the system:
      • The parabola opens upward.
      • The vertex is at point (negative 1.7 comma 0).
      • The parabola passes through the following points:
        • (negative 2.7 comma 1)
        • (negative 1.7 comma 0)
        • (negative 0.7 comma 1)

Show Answer Correct Answer: A

Choice A is correct. The graph of a quadratic equation in the form y=x2+c has its vertex at (0,c). The first equation in the given system of equations is y=x2+1.7, so the graph of this quadratic equation has its vertex at (0,1.7). The graph of a linear equation of the form y=b-x has a slope of -1 and a y-intercept at (0,b). The second equation in the given system of equations is y=1.7-x, so the graph of this linear equation has a slope of -1 and a y-intercept at (0,1.7). Of the choices, only choice A has the graph of a quadratic equation with its vertex at (0,1.7) and the graph of a linear equation with a slope of -1 and a y-intercept at (0,1.7).

Choice B is incorrect. This graph represents a system in which the second equation is y=1.7+x, not y=1.7-x.

Choice C is incorrect. This graph represents a system in which the first equation is y=(x+1.7)2, not y=x2+1.7.

Choice D is incorrect. This graph represents a system in which the first equation is y=(x+1.7)2, not y=x2+1.7, and the second equation is y=1.7+x, not y=1.7-x.

Question 167 167 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

x squared, minus a, x, plus 12, equals 0

In the equation above, a is a constant and a, is greater than 0. If the equation has two integer solutions, what is a possible value of a ?

Show Answer

The correct answer is either 7, 8, or 13. Since the given equation has two integer solutions, the expression on the left-hand side of this equation can be factored as open parenthesis, x plus c, close parenthesis, times, open parenthesis, x plus d, close parenthesis, where c and d are also integers. The product of c and d must equal the constant term of the original quadratic expression, which is 12. Additionally, the sum of c and d must be a negative number since it’s given that a, is greater than 0, but the sign preceding a in the given equation is negative. The possible pairs of values for c and d that satisfy both of these conditions are negative 4 and negative 3, negative 6 and negative 2, and negative 12 and negative 1. Since the value of negative a is the sum of c and d, the possible values of negative a are negative 4 plus negative 3, equals negative 7, negative 6 plus negative 2, equals negative 8, and negative 12 plus negative 1, equals negative 13. It follows that the possible values of a are 7, 8, and 13. Note that 7, 8, and 13 are examples of ways to enter a correct answer.

Question 168 168 of 479 selected Nonlinear Functions M

h(t)=-16t2+b

The function h estimates an object’s height, in feet, above the ground t seconds after the object is dropped, where b is a constant. The function estimates that the object is 3,364 feet above the ground when it is dropped at t = 0 . Approximately how many seconds after being dropped does the function estimate the object will hit the ground?

  1. 7.25

  2. 14.50

  3. 105.13

  4. 210.25

Show Answer Correct Answer: B

Choice B is correct. It's given that the function h estimates that the object is 3,364 feet above the ground when it's dropped at t = 0 . Substituting 3,364 for h(t) and 0 for t in the function h yields 3,364=-16(0)2+b, or 3,364 = b . Substituting 3,364 for b in the function h yields h(t)=-16t2+3,364. When the object hits the ground, its height will be 0 feet above the ground. Substituting 0 for h(t) in h(t)=-16t2+3,364 yields 0 = - 16 t 2 + 3,364 . Adding 16 t 2 to each side of this equation yields 16 t 2 = 3,364 . Dividing each side of this equation by 16 yields t 2 = 210.25 . Since the object will hit the ground at a positive number of seconds after it's dropped, the value of t can be found by taking the positive square root of each side of this equation, which yields t=14.50. It follows that the function estimates the object will hit the ground approximately 14.50 seconds after being dropped.

Choice A is incorrect. The function estimates that 7.25 seconds after being dropped, the object's height will be -16(7.25)2+3,364 feet, or 2,523 feet, above the ground.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 169 169 of 479 selected Nonlinear Functions M

From 2005 through 2014, the number of music CDs sold in the United States declined each year by approximately 15% of the number sold the preceding year. In 2005, approximately 600 million CDs were sold in the United States. Of the following, which best models C, the number of millions of CDs sold in the United States, t years after 2005?

  1. C equals, 600 times, open parenthesis, 0 point 1 5, close parenthesis, raised to the t power

  2. C equals, 600 times, open parenthesis, 0 point 8 5, close parenthesis, raised to the t power

  3. C equals, 600 times, open parenthesis, 1 point 1 5, close parenthesis, raised to the t power

  4. C equals, 600 times, open parenthesis, 1 point 8 5, close parenthesis, raised to the t power

Show Answer Correct Answer: B

Choice B is correct. A model for a quantity C that decreases by a certain percentage per time period t is an exponential equation in the form C equals, I times, open parenthesis, 1 minus, the fraction r over 100, close parenthesis, to the t power, where I is the initial value at time t equals 0 for r% annual decline. It’s given that C is the number of millions of CDs sold in the United States and that t is the number of years after 2005. It’s also given that 600 million CDs were sold at time t equals 0, so I equals 600. This number declines by 15% per year, so r equals 15. Substituting these values into the equation produces C equals, 600 times, open parenthesis, 1 minus the fraction 15 over 100, close parenthesis, to the t power  , or C equals, 600 times, open parenthesis, 0 point 8 5, close parenthesis, to the t power.

Choice A is incorrect and may result from errors made when representing the percent decline. Choices C and D are incorrect. These equations model exponential increases in CD sales, not exponential decreases.

 

Question 170 170 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

The square root of 2 x plus 6, end root, plus 4 equals, x plus 3

What is the solution set of the equation above?

  1. set consisting of negative 1

  2. set consisting of 5

  3. set consisting of negative 1 and 5

  4. set consisting of 0, negative 1, and 5

Show Answer Correct Answer: B

Choice B is correct. Subtracting 4 from both sides of the square root of 2 x plus 6, end root, plus 4, equals, x plus 3 isolates the radical expression on the left side of the equation as follows: the square root of 2 x plus 6, end root, equals, x minus 1. Squaring both sides of the square root of 2 x plus 6, end root, equals, x minus 1 yields 2 x plus 6, equals x squared, minus 2 x, plus 1. This equation can be rewritten as a quadratic equation in standard form: x squared, minus 4 x, minus 5, equals 0. One way to solve this quadratic equation is to factor the expression x squared, minus 4 x, minus 5 by identifying two numbers with a sum of negative 4 and a product of negative 5. These numbers are negative 5 and 1. So the quadratic equation can be factored as open parenthesis, x minus 5, close parenthesis, times, open parenthesis, x plus 1, close parenthesis, equals 0. It follows that 5 and negative 1 are the solutions to the quadratic equation. However, the solutions must be verified by checking whether 5 andnegative 1 satisfy the original equation, the square root of 2 x plus 6, end root, plus 4, equals, x plus 3. When x equals negative 1, the original equation gives the square root of 2 times negative 1, plus 6, end root, plus 4, equals, negative 1 plus 3, or 6 equals 2, which is false. Therefore, negative 1 does not satisfy the original equation. When x equals 5, the original equation gives the square root of 2 times 5, plus 6, end root, plus 4, equals, 5 plus 3, or 8 equals 8, which is true. Therefore, x equals 5 is the only solution to the original equation, and so the solution set is 5.

Choices A, C, and D are incorrect because each of these sets contains at least one value that results in a false statement when substituted into the given equation. For instance, in choice D, when 0 is substituted for x into the given equation, the result is the square root of 2 times 0, plus 6, plus 4, end root, equals, 0 plus 3, or the square root of 6, end root, plus 4, equals 3. This is not a true statement, so 0 is not a solution to the given equation.

 

Question 171 171 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H
The figure presents the graph of a curve in the xy-plane. The curve is labeled y equals f of x. The numbers 0 through 9 are indicated on the x-axis. The numbers 0 through 12, in increments of 2, are indicated on the y-axis. The curve begins at the point with coordinates 0 comma 2 and moves upward and to the right reaching a maximum at the point with coordinates 4 comma 10. It turns and moves downward and to the right, ending at 8 point 5 on the x-axis

The graph of the function f, defined by f of x equals, negative one-half times, open parenthesis, x minus 4, close parenthesis, squared, plus 10, is shown in the xy-plane above. If the function g (not shown) is defined by g of x equals, negative x plus 10, what is one possible value of a such that f of a equals, g of a ?

Show Answer

The correct answer is either 2 or 8. Substituting x equals a in the definitions for f and g gives f of a, equals, negative one half times, open parenthesis, a, minus 4, close parenthesis, squared, plus 10 and g of a, equals, negative a, plus 10, respectively. If f of a, equals, g of a, then negative one half times, open parenthesis, a, minus 4, close parenthesis, squared, plus 10, equals, negative a, plus 10. Subtracting 10 from both sides of this equation gives negative one half times, open parenthesis, a, minus 4, close parenthesis, squared, equals negative a. Multiplying both sides by negative 2 gives open parenthesis, a, minus 4, close parenthesis, squared, equals 2 a. Expanding open parenthesis, a, minus 4, close parenthesis, squared gives a, squared, minus 8 a, plus 16, equals 2 a. Combining the like terms on one side of the equation gives a, squared, minus 10 a, plus 16, equals 0. One way to solve this equation is to factor a, squared, minus 10 a, plus 16 by identifying two numbers with a sum of negative 10 and a product of 16. These numbers are negative 2 and negative 8, so the quadratic equation can be factored as open parenthesis, a, minus 2, close parenthesis, times, open parenthesis, a, minus 8, close parenthesis, equals 0. Therefore, the possible values of a are either 2 or 8. Note that 2 and 8 are examples of ways to enter a correct answer.

Alternate approach: Graphically, the condition f of a, equals, g of a implies the graphs of the functions y equals f of x and y equals g of x intersect at x equals a. The graph y equals f of x is given, and the graph of y equals g of x may be sketched as a line with y-intercept 10 and a slope of negative 1 (taking care to note the different scales on each axis). These two graphs intersect at x equals 2 and x equals 8.

Question 172 172 of 479 selected Equivalent Expressions M

The expression 24 6 x + 42 is equivalent to 4 x + b , where b is a constant and x>0. What is the value of b?

  1. 7

  2. 10

  3. 24

  4. 252

Show Answer Correct Answer: A

Choice A is correct. Since the given expressions are equivalent and the numerator of the second expression is 16 of the numerator of the first expression, the denominator of the second expression must also be 16 of the denominator of the first expression. By the distributive property,  16(6x+42) is equivalent to 16(6x)+16(42), or x+7. Therefore, the value of b is 7 .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 173 173 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

x 2 - 34 x + c = 0

In the given equation, c is a constant. The equation has no real solutions if c>n. What is the least possible value of n ?

Show Answer Correct Answer: 289

The correct answer is 289 . A quadratic equation of the form ax2+bx+c=0, where a , b , and c are constants, has no real solutions when the value of the discriminant, b2-4ac, is less than 0 . In the given equation, x2-34x+c=0, a=1 and b=-34. Therefore, the discriminant of the given equation can be expressed as (-34)2-4(1)(c), or 1,156-4c. It follows that the given equation has no real solutions when 1,156-4c<0. Adding 4 c to both sides of this inequality yields 1,156<4c. Dividing both sides of this inequality by 4 yields 289<c, or c>289. It's given that the equation x2-34x+c=0 has no real solutions when c>n. Therefore, the least possible value of n is 289 .

Question 174 174 of 479 selected Equivalent Expressions H

Which of the following expressions has a factor of x+2b, where b is a positive integer constant?

  1. 3 x 2 + 7 x + 14 b

  2. 3 x 2 + 28 x + 14 b

  3. 3 x 2 + 42 x + 14 b

  4. 3 x 2 + 49 x + 14 b

Show Answer Correct Answer: D

Choice D is correct. Since each choice has a term of 3x2, which can be written as (3x)(x), and each choice has a term of 14b, which can be written as (7)(2b), the expression that has a factor of x+2b, where b is a positive integer constant, can be represented as (3x+7)(x+2b). Using the distributive property of multiplication, this expression is equivalent to 3x(x+2b)+7(x+2b), or 3x2+6xb+7x+14b. Combining the x-terms in this expression yields 3x2+(7+6b)x+14b. It follows that the coefficient of the x-term is equal to 7+6b. Thus, from the given choices, 7+6b must be equal to 7 , 28 , 42 , or 49 . Therefore, 6b must be equal to 0 , 21 , 35 , or 42 , respectively, and b must be equal to 06216356, or 426, respectively. Of these four values of b , only 426, or 7 , is a positive integer. It follows that 7+6b must be equal to 49 because this is the only choice for which the value of b is a positive integer constant. Therefore, the expression that has a factor of x+2b is 3x2+49x+14b.

Choice A is incorrect. If this expression has a factor of x+2b, then the value of b is 0 , which isn't positive.

Choice B is incorrect. If this expression has a factor of x+2b, then the value of b is 216, which isn't an integer.

Choice C is incorrect. If this expression has a factor of x+2b, then the value of b is 35 6 , which isn't an integer.

Question 175 175 of 479 selected Nonlinear Functions M

  • The parabola opens downward.
  • The vertex is at point (2 comma 28).
  • The parabola passes through the following points:
    • (0 comma 8.4)
    • (2 comma 28)
    • (4 comma 8.4)

An object was launched upward from a platform. The graph shown models the height above ground, y , in meters, of the object x seconds after it was launched. For which of the following intervals of time was the height of the object increasing for the entire interval?

  1. From x = 0 to x = 2

  2. From x = 0 to x = 4

  3. From x = 2 to x = 3

  4. From x = 3 to x = 4

Show Answer Correct Answer: A

Choice A is correct. It's given that the variable y represents the height, in meters, of the object above the ground. The graph shows that the height of the object was increasing from x = 0 to x = 2 , and decreasing from x = 2 to x = 4 . Therefore, the height of the object was increasing for the entire interval of time from x = 0 to x = 2 .

Choice B is incorrect. The height of the object wasn't increasing for this entire interval of time, as it was decreasing from x = 2 to x = 4 .

Choice C is incorrect. The height of the object was decreasing, not increasing, for this entire interval of time.

Choice D is incorrect. The height of the object was decreasing, not increasing, for this entire interval of time.

Question 176 176 of 479 selected Nonlinear Functions H

The population P of a certain city y years after the last census is modeled by the equation below, where r is a constant and P subscript 0 is the population when y equals 0.

P equals, P subscript 0, times, open parenthesis, 1 plus r, close parenthesis, to the power y

If during this time the population of the city decreases by a fixed percent each year, which of the following must be true?

  1. r is less than negative 1
  2. negative 1 is less than r, which is less than 0
  3. 0 is less than r, which is less than 1
  4. r is greater than 1
Show Answer Correct Answer: B

Choice B is correct. The term (1 + r) represents a percent change. Since the population is decreasing, the percent change must be between 0% and 100%. When the percent change is expressed as a decimal rather than as a percent, the percentage change must be between 0 and 1. Because (1 + r) represents percent change, this can be expressed as 0 < 1 + r < 1. Subtracting 1 from all three terms of this compound inequality results in –1 < r < 0.

Choice A is incorrect. If r < –1, then after 1 year, the population P would be a negative value, which is not possible. Choices C and D are incorrect. For any value of r > 0, 1 + r > 1, and the exponential function models growth for positive values of the exponent. This contradicts the given information that the population is decreasing.

Question 177 177 of 479 selected Equivalent Expressions E

Which expression is equivalent to parenthesis, 2 x squared minus 4, close parenthesis, minus, parenthesis, negative 3 x squared, plus 2 x, minus 7, close parenthesis ?

  1. 5 x squared minus 2 x plus 3

  2. 5 x squared plus 2 x minus 3

  3. negative x squared minus 2 x minus 11

  4. negative x squared plus 2 x minus 11

Show Answer Correct Answer: A

Choice A is correct. The given expression open parenthesis, 2 x squared, minus 4, close parenthesis, minus, open parenthesis, negative 3, x squared, plus 2 x, minus 7, close parenthesis can be rewritten as 2 x squared, minus 4, plus 3 x squared, minus 2 x, plus 7. Combining like terms yields 5 x squared, minus 2 x, plus 3.

Choices B, C, and D are incorrect and may be the result of errors when applying the distributive property.

 

Question 178 178 of 479 selected Nonlinear Functions E

  • For the first curve: 
    • Moving from left to right:
      • The curve passes from quadrant 2 to quadrant 1.
      • In quadrant 2, the curve trends up gradually to point (0 comma 3).
      • In quadrant 1, the curve trends up sharply.
    • As x increases, the curve approaches the line x equals 4.
    • As x decreases, the curve approaches the line y equals 1.
    • The curve passes through the following points:
      • (negative 3 comma 2)
      • (0 comma 3)
  • For the second curve: 
    • Moving from left to right:
      • The curve is in quadrant 4.
      • In quadrant 4, the curve trends up sharply.
    • As x increases, the curve approaches the line y equals 1.
    • As x decreases, the curve approaches the line x equals 4.
    • The curve passes through the following point:
      • (6 comma negative 4)

The graph of y=f(x) is shown in the xy-plane. What is the value of f(0)?

  1. -3

  2. 0

  3. 3 5

  4. 3

Show Answer Correct Answer: D

Choice D is correct. Because the graph of y=f(x) is shown, the value of f(0) is the value of y on the graph that corresponds with x = 0 . When x = 0 , the corresponding value of y is 3 . Therefore, the value of f(0) is 3 .

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Question 179 179 of 479 selected Nonlinear Functions E

The function f is defined by f(x)=10x2-32x-152. What is the value of f(0)?

  1. -152

  2. -32

  3. 0

  4. 10

Show Answer Correct Answer: A

Choice A is correct. The value of f(0) is the value of f(x) when x = 0 . The function f is defined by f(x)=10x2-32x-152. Substituting 0 for x in this equation yields f(0)=10(0)2-32(0)-152. This equation can be rewritten as f(0)=10(0)-0-152, or f(0)=-152. Therefore, the value of f(0) is - 152 .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 180 180 of 479 selected Nonlinear Functions M

  • Moving from left to right:
    • The curve is in quadrant 1.
    • From point (0 comma 9), the curve trends down sharply.
    • From point (5 comma 1), the curve trends down gradually.

The graph gives the estimated number of catalogs y , in thousands, a company sent to its customers at the end of each year, where x represents the number of years since the end of 1992 , where 0x10. Which statement is the best interpretation of the y-intercept in this context?

  1. The estimated total number of catalogs the company sent to its customers during the first 10 years was 9,000 .

  2. The estimated total number of catalogs the company sent to its customers from the end of 1992 to the end of 2002 was 90 .

  3. The estimated number of catalogs the company sent to its customers at the end of 1992 was 9 .

  4. The estimated number of catalogs the company sent to its customers at the end of 1992 was 9,000 .

Show Answer Correct Answer: D

Choice D is correct. The y-intercept of the graph is the point at which the graph crosses the y-axis, or the point for which the value of x is 0 . Therefore, the y-intercept of the given graph is the point (0,9). It's given that x represents the number of years since the end of 1992. Therefore, x=0 represents 0 years since the end of 1992, which is the same as the end of 1992. It's also given that y represents the estimated number of catalogs, in thousands, that the company sent to its customers at the end of the year. Therefore, y=9 represents 9,000 catalogs. It follows that the y-intercept (0,9) means that the estimated number of catalogs the company sent to its customers at the end of 1992 was 9,000

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 181 181 of 479 selected Nonlinear Functions H

‒10‒8‒6‒4‒2246810x‒10‒8‒6‒4‒224yO
  • Moving from left to right, the curve passes from quadrant 3 to quadrant 4.
  • As x decreases, the curve approaches line y equals negative 7.
  • The curve passes through the following approximate points:
    • (negative 1 comma negative 7.3)
    • (0 comma negative 7.9)

The graph of y=f(x) is shown, where f(x)=abx+c, and a , b , and c are constants. For how many values of x does f(x)=0?

  1. Three

  2. Two

  3. One

  4. Zero

Show Answer Correct Answer: D

Choice D is correct. Each point (x,y) on the graph of y=f(x) in the xy-plane gives a value of x and its corresponding value of f(x), or y. For any value of x for which f(x)=0, there is a corresponding point on the graph of y=f(x) with a y-coordinate of 0. A point with a y-coordinate of 0 is a point where the graph intersects the x-axis. It's given that f(x)=abx+c, where a, b, and c are constants. In the xy-plane, the graph of an equation of this form will lie entirely either above or below the horizontal line defined by y=c. The part of the graph of y=f(x) shown lies entirely below the horizontal line defined by y=-7, and thus the entire graph of y=f(x) must lie below the line defined by y=-7. It follows that the graph of y=f(x) will never intersect the x-axis. Therefore, there are zero values of x for which f(x)=0.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Question 182 182 of 479 selected Nonlinear Functions M

f(x)=(x+6)(x+5)(x-4)

The function f is given. Which table of values represents y=f(x)-3?

  1. x y
    -6 -9
    -5 -8
    4 1
  2. x y
    -6 -3
    -5 -3
    4 -3
  3. x y
    -6 -3
    -5 -2
    4 7
  4. x y
    -6 3
    -5 3
    4 3
Show Answer Correct Answer: B

Choice B is correct. It’s given that f(x)=(x+6)(x+5)(x-4) and y=f(x)-3. Substituting (x+6)(x+5)(x-4) for f(x) in the equation y=f(x)-3 yields y=(x+6)(x+5)(x-4)-3. Substituting -6 for x in this equation yields y=(-6+6)(-6+5)(-6-4)-3, or y = -3 . Substituting -5 for x in the equation y=(x+6)(x+5)(x-4)-3 yields y=(-5+6)(-5+5)(-5-4)-3, or y = -3 . Substituting 4 for x in the equation y=(x+6)(x+5)(x-4)-3 yields y=(4+6)(4+5)(4-4)-3, or y = -3 . Therefore, when x = -6 then y = -3 , when x = -5 then y = -3 , and when x = 4 then y = -3 . Thus, the table of values in choice B represents y=f(x)-3.

Choice A is incorrect. This table represents y = x - 3 rather than y=f(x)-3.

Choice C is incorrect. This table represents y = x + 3 rather than y=f(x)-3.

Choice D is incorrect. This table represents y=f(x)+3 rather than y=f(x)-3.

Question 183 183 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

| x - 5 | = 10

What is one possible solution to the given equation?

Show Answer Correct Answer: 15, -5

The correct answer is 15 or -5 . By the definition of absolute value, if |x-5|=10, then x-5=10 or x-5=-10. Adding 5 to both sides of the first equation yields x=15. Adding 5 to both sides of the second equation yields x=-5. Thus, the given equation has two possible solutions, 15 and -5 . Note that 15 and -5 are examples of ways to enter a correct answer.

Question 184 184 of 479 selected Equivalent Expressions H

Which expression is equivalent to 42ak+42ak, where k>0?

  1. 84ak

  2. 84ak2k

  3. 42a(k+1)k

  4. 42a(k2+1)k

Show Answer Correct Answer: D

Choice D is correct. Two fractions can be added together when they have a common denominator. Since k>0, multiplying the second term in the given expression by kk yields (42ak)kk, which is equivalent to 42ak2k. Therefore, the expression 42ak+42ak can be written as 42ak+42ak2k which is equivalent to 42a+42ak2k. Since each term in the numerator of this expression has a factor of 42 a , the expression 42a+42ak2k can be rewritten as 42a(1)+42a(k2)k, or 42a(1+k2)k, which is equivalent to  42a(k2+1)k.

Choice A is incorrect. This expression is equivalent to 42ak + 42ak.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This expression is equivalent to 42ak + 42a.

Question 185 185 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

x = 3

y=(15-x)2

A solution to the given system of equations is (x,y). What is the value of x y ?

  1. 432

  2. 54

  3. 45

  4. 18

Show Answer Correct Answer: A

Choice A is correct. The first equation in the given system of equations is x=3. Substituting 3 for x in the second equation in the given system of equations yields y=(15-3)2, or y=144. Substituting 3 for x and 144 for y in the expression x y yields (3)(144), or 432 . Therefore, the value of x y is 432 .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 186 186 of 479 selected Nonlinear Functions E

The function f is defined by f(x)=x2+x+71. What is the value of f(2)?

Show Answer Correct Answer: 77

The correct answer is 77 . It’s given that the function f is defined by f(x)=x2+x+71. Substituting 2 for x in function f yields f(2)=(2)2+2+71, which is equivalent to f(2)=4+2+71, or f(2)=77. Therefore, the value of f(2) is 77 .

Question 187 187 of 479 selected Nonlinear Functions M

The product of two positive integers is 546 . If the first integer is 11 greater than twice the second integer, what is the smaller of the two integers?

  1. 7

  2. 14

  3. 39

  4. 78

Show Answer Correct Answer: B

Choice B is correct. Let x be the first integer and let y be the second integer. If the first integer is 11 greater than twice the second integer, then x = 2 y + 11 . If the product of the two integers is 546 , then xy=546. Substituting 2 y + 11 for x in this equation results in (2y+11)y=546. Distributing the y to both terms in the parentheses results in 2 y 2 + 11 y = 546 . Subtracting 546 from both sides of this equation results in 2 y 2 + 11 y - 546 = 0 . The left-hand side of this equation can be factored by finding two values whose product is 2(-546), or -1,092 , and whose sum is 11 . The two values whose product is -1,092 and whose sum is 11 are 39 and -28 . Thus, the equation 2 y 2 + 11 y - 546 = 0 can be rewritten as 2y2+28y-39y-546=0, which is equivalent to 2y(y-14)+39(y-14)=0, or (2y+39)(y-14)=0. By the zero product property, it follows that 2 y + 39 = 0 and y - 14 = 0 . Subtracting 39 from both sides of the equation 2 y + 39 = 0 yields 2 y = -39 . Dividing both sides of this equation by 2 yields y = - 39 2 . Since y is a positive integer, the value of y is not - 39 2 . Adding 14 to both sides of the equation y - 14 = 0 yields y = 14 . Substituting 14 for y in the equation xy=546 yields x(14)=546. Dividing both sides of this equation by 14 results in x = 39 . Therefore, the two integers are 14 and 39 , so the smaller of the two integers is 14 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the larger of the two integers.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 188 188 of 479 selected Nonlinear Functions H

f(x)=5,470(0.64)x12

The function f gives the value, in dollars, of a certain piece of equipment after x months of use. If the value of the equipment decreases each year by p% of its value the preceding year, what is the value of p ?

  1. 4

  2. 5

  3. 36

  4. 64

Show Answer Correct Answer: C

Choice C is correct. For a function of the form f(x)=a(r)xk, where a , r , and k are constants and r<1, the value of f(x) decreases by 100(1-r)% for every increase of x by k . In the given function, a = 5,470 , r = 0.64 , and k = 12 . Therefore, for the given function, the value of f(x) decreases by 100(1-0.64)%, or 36%, for every increase of x by 12 . Since f(x) represents the value, in dollars, of the equipment after x months of use, it follows that the value of the equipment decreases every 12 months by 36% of its value the preceding 12 months. Since there are 12 months in a year, the value of the equipment decreases each year by 36% of its value the preceding year. Thus, the value of p is 36 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 189 189 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

p + 34 = q + r

The given equation relates the variables p , q , and r . Which equation correctly expresses p in terms of q and r

  1. p=q+r+34

  2. p=q+r-34

  3. p=-q-r+34

  4. p=-q-r-34

Show Answer Correct Answer: B

Choice B is correct. Subtracting 34 from each side of the given equation yields p = q + r - 34 . Thus, the equation p = q + r - 34 correctly expresses p in terms of q and r .

Choice A is incorrect. This equation can be rewritten as p - 34 = q + r .

Choice C is incorrect. This equation can be rewritten as p - 34 = - q - r .

Choice D is incorrect. This equation can be rewritten as p + 34 = - q - r .

Question 190 190 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

- 16 x 2 - 8 x + c = 0

In the given equation, c is a constant. The equation has exactly one solution. What is the value of c ?

Show Answer Correct Answer: -1

The correct answer is -1 . A quadratic equation in the form ax2+bx+c=0, where a , b , and c are constants, has exactly one solution when its discriminant, b2-4ac, is equal to 0 . In the given equation, -16x2-8x+c=0, a = -16 and b = -8 . Substituting -16 for a and -8 for b in b2-4ac yields (-8)2-4(-16)(c), or 64+64c. Since the given equation has exactly one solution, 64+64c=0. Subtracting 64 from both sides of this equation yields 64c=-64. Dividing both sides of this equation by 64 yields c = -1 . Therefore, the value of c is -1 .

Question 191 191 of 479 selected Equivalent Expressions H

The expression the fraction with numerator x to the power negative 2, end power, times y to the power one-half, and denominator x to the power one-third, end power, times y to the power negative 1, end fraction, where x is greater than 1 and y is greater than 1 , is equivalent to which of the following?

  1. The fraction with numerator the square root of y and denominator the cube root of x squared

  2. The fraction with numerator y times the square root of y and denominator the cube root of x squared

  3. The fraction with numerator y times the square root of y and denominator x times the square root of x

  4. The fraction with numerator y times the square root of y and denominator x squared times the cube root of x

Show Answer Correct Answer: D

Choice D is correct. For x is greater than 1 and y is greater than 1 , x to the one third power and y to the one half power are equivalent to the cube root of x and the square root of y, respectively. Also, x to the negative 2 power and y to the negative 1 power are equivalent to the fraction 1 over x squared, end fraction and 1 over y, respectively. Therefore, the given expression can be rewritten as the fraction with numerator, y times the square root of y, and denominator x squared, times the cube root of x, end fraction.

Choices A, B, and C are incorrect because these choices are not equivalent to the given expression for x is greater than 1 and y is greater than 1.

For example, for x equals 2 and y equals 2, the value of the given expression is 2 to the negative five sixths power; the values of the choices, however, are 2 to the negative one third power, 2 to the five sixths power, and 1, respectively.

 

Question 192 192 of 479 selected Nonlinear Functions M

In the xy-plane, the y-coordinate of the y-intercept of the graph of the function f is c. Which of the following must be equal to c ?

  1. f of 0

  2. f of 1

  3. f of 2

  4. f of 3

Show Answer Correct Answer: A

Choice A is correct. A y-intercept is the point in the xy-plane where the graph of the function crosses the y-axis, which is where x equals 0. It’s given that the y-coordinate of the y-intercept of the graph of function f is c. It follows that the coordinate pair representing the y-intercept must be the point with coordinates 0 comma c. Therefore, c must equal f of 0.

Choices B, C, and D are incorrect because f of 1, f of 2, and f of 3would represent the y-value of the coordinate where x equals 1, x equals 2, and x equals 3, respectively.

 

Question 193 193 of 479 selected Nonlinear Functions M

f(t)=500(0.5)t12

The function f models the intensity of an X-ray beam, in number of particles in the X-ray beam, t millimeters below the surface of a sample of iron. According to the model, what is the estimated number of particles in the X-ray beam when it is at the surface of the sample of iron?

  1. 500

  2. 12

  3. 5

  4. 2

Show Answer Correct Answer: A

Choice A is correct. It's given that the function f models the intensity of an X-ray beam, in number of particles in the X-ray beam, t millimeters below the surface of a sample of iron. When the X-ray beam is at the surface of the sample of iron, it is 0 millimeters below the surface, so the value of t is 0 . Substituting 0 for t in the function f(t)=500(0.5)t12 yields f(0)=500(0.5)012. Since any positive number raised to the power of 0 is equal to 1 , it follows that f(0)=500(1), or f(0)=500. Therefore, the estimated number of particles in the X-ray beam at the surface of the sample of iron is 500 .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 194 194 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

5 x 2 - 37 x - 24 = 0

What is the positive solution to the given equation?

  1. 3 5

  2. 3

  3. 8

  4. 37

Show Answer Correct Answer: C

Choice C is correct. The left-hand side of the given equation can be factored as (5x+3)(x-8). Therefore, the given equation, 5x2-37x-24=0, can be written as (5x+3)(x-8)=0. Applying the zero product property to this equation yields 5x+3=0 and x-8=0. Subtracting 3 from both sides of the equation 5x+3=0 yields 5x=-3. Dividing both sides of this equation by 5 yields x=-35. Adding 8 to both sides of the equation x-8=0 yields x=8. Therefore, the two solutions to the given equation, 5x2-37x-24=0, are -35 and 8 . It follows that 8 is the positive solution to the given equation.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 195 195 of 479 selected Equivalent Expressions H

The expression 4 x 2 + b x - 45 , where b is a constant, can be rewritten as (hx+k)(x+j), where h , k , and j are integer constants. Which of the following must be an integer?

  1. b h

  2. b k

  3. 45 h

  4. 45 k

Show Answer Correct Answer: D

Choice D is correct. It's given that 4x2+bx-45 can be rewritten as (hx+k)(x+j). The expression (hx+k)(x+j) can be rewritten as hx2+jhx+kx+kj, or hx2+(jh+k)x+kj. Therefore, hx2+(jh+k)x+kj is equivalent to 4x2+bx-45. It follows that kj=-45. Dividing each side of this equation by k yields j=-45k. Since j is an integer, - 45 k must be an integer. Therefore, 45 k must also be an integer.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 196 196 of 479 selected Nonlinear Functions M

y = 0.25 x 2 - 7.5 x + 90.25

The equation gives the estimated stock price y , in dollars, for a certain company x days after a new product launched, where 0x20. Which statement is the best interpretation of (x,y)=(1,83)  in this context?

  1. The company's estimated stock price increased $83 every day after the new product launched.

  2. The company's estimated stock price increased $1 every 83 days after the new product launched.

  3. 1 day after the new product launched, the company's estimated stock price is $83 .

  4. 83 days after the new product launched, the company's estimated stock price is $1.

Show Answer Correct Answer: C

Choice C is correct. In the given equation, x represents the number of days after a new product launched, where 0x20, and y represents the estimated stock price, in dollars, for a certain company. Therefore, the best interpretation of (x,y)=(1,83) in this context is that 1 day after the new product launched, the company's estimated stock price is $83.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.
 

Question 197 197 of 479 selected Nonlinear Functions M

On April 1, there were 233 views of an advertisement posted on a website. Every 2 days after April 1, the number of views of the advertisement had increased by 70 % of the number of views 2 days earlier. The function f gives the predicted number of views x days after April 1. Which equation defines f?

  1. f(x)=233(0.70)x2

  2. f(x)=233(0.70)2x

  3. f(x)=233(1.70)x2

  4. f(x)=233(1.70)2x

Show Answer Correct Answer: C

Choice C is correct. It’s given that on April 1, there were 233 views of the advertisement. It’s also given that every 2 days after April 1, the number of views of the advertisement had increased by 70% of the number of views 2 days earlier. This situation can be represented by an exponential function of the form f(x)=a(1+r100)xc, where a is the number of views on April 1 and every c days after April 1, the number of views had increased by r% of the number of views c days earlier. It follows that a=233, r=70, and c=2. Substituting 233 for a, 70 for r, and 2 for c in the equation f(x)=a(1+r100)xc yields f(x)=233(1+70100)x2, or f(x)=233(1.70)x2.

Choice A is incorrect. This function gives the predicted number of views for an advertisement for which every 2 days, the number of views was 70%, rather than increased by 70%, of the number of views 2 days earlier.

Choice B is incorrect. This function gives the predicted number of views for an advertisement for which every 12 days, the number of views was 70% of the number of views 12 days earlier, rather than an advertisement for which every 2 days, the number of views had increased by 70% of the number of views 2 days earlier.

Choice D is incorrect. This function gives the predicted number of views for an advertisement for which every 12 days, rather than every 2 days, the number of views had increased by 70% of the number of views 12 days earlier, rather than 2 days earlier.

Question 198 198 of 479 selected Nonlinear Functions E

  • The curve is in quadrant 1.
  • The curve trends gradually up from left to right. 
  • The curve begins at the point (0 comma 20,000).
  • The curve passes through the following points:
    • (0 comma 20,000)
    • approximately (5 comma 26,263)
    • approximately (10 comma 34,488)

The graph shown models the number of residents of a certain city x years after 2010. How many residents does this model estimate the city had in 2010?

  1. 0

  2. 2,000

  3. 20,000

  4. 25,000

Show Answer Correct Answer: C

Choice C is correct. It's given that x represents years after 2010. Therefore, 2010 is represented by x = 0 . On the model shown, the point with an x-coordinate of 0 has a y-coordinate of 20,000 . Thus, the model estimates that in 2010, the city had 20,000 residents.

Choice A is incorrect. This is the value of x that represents the year 2010.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is approximately the number of residents the model estimates the city had in 2014, not 2010.

Question 199 199 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

( x - 1 ) 2 = -4

How many distinct real solutions does the given equation have?

  1. Exactly one

  2. Exactly two

  3. Infinitely many

  4. Zero

Show Answer Correct Answer: D

Choice D is correct. Any quantity that is positive or negative in value has a positive value when squared. Therefore, the left-hand side of the given equation is either positive or zero for any value of x . Since the right-hand side of the given equation is negative, there is no value of x for which the given equation is true. Thus, the number of distinct real solutions for the given equation is zero.

Choices A, B, and C are incorrect and may result from conceptual errors.

Question 200 200 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

x2=(22)(22)

What is the positive solution to the given equation?

Show Answer Correct Answer: 22

The correct answer is 22 . The given equation, x2=(22)(22), is equivalent to x2=(22)2. Taking the square root of each side of this equation yields x=±22. Thus, the positive solution to the given equation is 22 .

Question 201 201 of 479 selected Equivalent Expressions E

Which expression is a factor of 2 x 2 + 38 x + 10 ?

  1. 2

  2. 5 x

  3. 38 x

  4. 2 x 2

Show Answer Correct Answer: A

Choice A is correct. Since 2 is a common factor of each of the terms in the given expression, the expression can be rewritten as 2(x2+19x+5). Therefore, the factors of the given expression are 2 and x 2 + 19 x + 5 . Of these two factors, only 2 is listed as a choice.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is a term of the given expression, not a factor of the given expression.

Choice D is incorrect. This is a term of the given expression, not a factor of the given expression.

Question 202 202 of 479 selected Equivalent Expressions H

f of x equals, x cubed minus, 9 x, and, g of x equals, x squared minus, 2 x minus 3

Which of the following expressions is equivalent to the fraction f of x over g of x, for x is greater than 3 ?

  1. the fraction with numerator 1, and denominator x plus 1, end fraction

  2. the fraction with numerator x plus 3, and denominator x plus 1, end fraction

  3. the fraction with numerator x times, open parenthesis, x minus 3, close parenthesis, and denominator x plus 1, end fraction

  4. the fraction with numerator x times, open parenthesis, x plus 3, close parenthesis, and denominator x plus 1, end fraction

Show Answer Correct Answer: D

Choice D is correct. Since x cubed, minus 9 x, equals, x times, open parenthesis, x plus 3, close parenthesis, times, open parenthesis, x minus 3, close parenthesis and x squared, minus 2 x, minus 3, equals, open parenthesis, x plus 1, close parenthesis, times, open parenthesis, x minus 3, close parenthesis, the fractionf of x, over g of x can be written as the fraction with numerator x times, open parenthesis, x plus 3, close parenthesis, times, open parenthesis, x minus 3, close parenthesis, and denominator, open parenthesis, x plus 1, close parenthesis, times, open parenthesis, x minus 3, close parenthesis, end fraction. It is given that x is greater than 3, so the common factor x minus 3 is not equal to 0. Therefore, the fraction can be further simplified to the fraction with numerator x times, open parenthesis, x plus 3, close parenthesis, and denominator x plus 1, end fraction.

Choice A is incorrect. The expression the fraction 1 over, x plus 1, end fraction is not equivalent to the fraction f of x, over g of x because at x equals 0, the fraction 1 over, x plus 1, end fraction as a value of 1 and the fraction f of x, over g of x has a value of 0.

Choice B is incorrect and results from omitting the factor x in the factorization of f of x. Choice C is incorrect and may result from incorrectly factoring g of x as open parenthesis, x plus 1, close parenthesis, times, open parenthesis, x plus 3, close parenthesis instead of open parenthesis, x plus 1, close parenthesis, times, open parenthesis, x minus 3, close parenthesis.

 

Question 203 203 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

z 2 + 10 z - 24 = 0

What is one of the solutions to the given equation?

Show Answer Correct Answer: 2, -12

The correct answer is either 2 or -12 . The left-hand side of the given equation can be rewritten by factoring. The two values that multiply to -24 and add to 10 are 12 and -2 . It follows that the given equation can be rewritten as (z+12)(z-2)=0. Setting each factor equal to 0 yields two equations: z + 12 = 0 and z - 2 = 0 . Subtracting 12 from both sides of the equation z + 12 = 0 results in z = -12 . Adding 2 to both sides of the equation z - 2 = 0 results in z = 2 . Note that 2 and -12 are examples of ways to enter a correct answer.

Question 204 204 of 479 selected Nonlinear Functions E

  • Moving from left to right:
    • The curve passes from quadrant 3 to quadrant 4 to quadrant 1.
    • In quadrant 3, the curve trends up gradually to the y axis.
    • In quadrant 4, the curve trends up gradually to the point (6 comma 0).
    • In quadrant 1, the curve trends up sharply.
  • As x decreases, the curve approaches the line y equals negative 2.
  • The curve passes through the following points:
    • (5 comma negative four thirds)
    • (6 comma 0)

What is the x-coordinate of the x-intercept of the graph shown?

Show Answer Correct Answer: 6

The correct answer is 6 . An x-intercept of a graph is a point on the graph where it intersects the x-axis, or where the value of y is 0 . The graph shown intersects the x-axis at the point (6,0). Therefore, the x-coordinate of the x-intercept of the graph shown is 6 .

Question 205 205 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

y = 18

y = - 3 ( x - 18 ) 2 + 15

If the given equations are graphed in the xy-plane, at how many points do the graphs of the equations intersect?

  1. Exactly one

  2. Exactly two

  3. Infinitely many

  4. Zero

Show Answer Correct Answer: D

Choice D is correct. A point (x,y) is a solution to a system of equations if it lies on the graphs of both equations in the xy-plane. In other words, a solution to a system of equations is a point (x,y) at which the graphs intersect. It’s given that the first equation is y = 18 . Substituting 18 for y in the second equation yields 18 = - 3 ( x - 18 ) 2 + 15 . Subtracting 15 from each side of this equation yields 3 = - 3 ( x - 18 ) 2 . Dividing each side of this equation by -3 yields -1 = ( x - 18 ) 2 . Since the square of a real number is at least 0 , this equation can't have any real solutions. Therefore, the graphs of the equations intersect at zero points.

Alternate approach: The graph of the second equation is a parabola that opens downward and has a vertex at (18,15). Therefore, the maximum value of this parabola occurs when y = 15 . The graph of the first equation is a horizontal line at 18 on the y-axis, or y = 18 . Since 18 is greater than 15 , or the horizontal line is above the vertex of the parabola, the graphs of these equations intersect at zero points.

Choice A is incorrect. The graph of y = 15 , not y = 18 , and the graph of the second equation intersect at exactly one point.

Choice B is incorrect. The graph of any horizontal line such that the value of y is less than 15 , not greater than 15 , and the graph of the second equation intersect at exactly two points.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 206 206 of 479 selected Nonlinear Functions M
f of x equals, x cubed, plus 3 x squared, minus 6 x, minus 1

For the function f defined above, what is the value of f of negative 1?

  1. negative 11​​​​​​​

  2. negative 7​​​​​​​

  3. 7

  4. 11

Show Answer Correct Answer: C

Choice C is correct. Substituting negative 1 for x in the given function f gives f of negative 1 equals, open parenthesis, negative 1, close parenthesis, cubed, plus 3 times, open parenthesis, negative 1, close parenthesis, squared, minus, 6 times negative 1, minus 1, which simplifies to f of negative 1 equals, negative 1 plus, 3 times 1, minus, 6 times negative 1, minus 1. This further simplifies to f of negative 1 equals, negative 1 plus 3, plus 6, minus 1, or f of negative 1, equals 7.

Choice A is incorrect and may result from correctly substituting negative 1 for x in the function but incorrectly simplifying the resulting expression to f of negative 1 equals, negative 1 minus 3, minus 6, minus 1, or negative 11. Choice B is incorrect and may result from arithmetic errors. Choice D is incorrect and may result from correctly substituting negative 1 for x in the function but incorrectly simplifying the expression to f of negative 1 equals, 1 plus 3, plus 6, plus 1, or 11.

 

Question 207 207 of 479 selected Nonlinear Functions M

y equals, 4 times, open parenthesis, 2 to the x power, close parenthesis

Which of the following is the graph in the xy-plane of the given equation?

  1.  

    The answer choice presents the graph of a curve in the x y plane, with the origin labeled O. The numbers negative 4 through 4, in increments of 1, are indicated on the x axis. The numbers negative 25 through 100, in increments of 25, are indicated on the y axis. The curve begins to the left of the y axis, high above the x-axis. As it moves rightward it goes downward, at first steeply, then more and more slowly as it gets closer and closer to the x axis. Eventually it goes downward toward the x axis so slowly it is almost horizontal. The curve passes through the point with coordinates negative 2 comma 32, and the point with coordinates negative 1 comma 8, and crosses the y-axis a little above the origin. It ends to the right of the y-axis, just above the x-axis.

     

  2.  

    The answer choice presents the graph of a curve in the x y plane, with the origin labeled O. The numbers negative 4 through 4, in increments of 1, are indicated on the x axis. The numbers negative 25 through 100, in increments of 25, are indicated on the y axis. The curve begins to the left of the y axis, high above the x axis. As it moves rightward it goes downward, at first steeply, then more and more slowly as it gets closer and closer to the x axis. Eventually it goes downward toward the x axis so slowly it is almost horizontal. It ends to the right of the y axis, just above the x axis. The curve passes through the point with coordinates negative 3 comma 32, the point with coordinates negative 2 comma 16, and the point with coordinates negative 1 comma 8. It crosses the y-axis a little above the origin and ends to the right of the y-axis, just above the x-axis.

     

  3.  

    The answer choice presents the graph of a curve in the x y plane, with the origin labeled O. The numbers negative 4 through 4, in increments of 1, are indicated on the x axis. The numbers negative 25 through 100, in increments of 25, are indicated on the y axis. The curve begins to the left of the y axis, just above the x axis. As it moves rightward, it goes upward, at first so slowly it is almost horizontal, then more and more quickly. The curve crosses the y-axis a little above the origin, and passes through the point with coordinates 1 comma 8 and the point with coordinates 2 comma 32. It ends to the right of the y-axis, high above the x-axis.

     

  4.  

    The answer choice presents the graph of a curve in the x y plane, with the origin labeled O. The numbers negative 4 through 4, in increments of 1, are indicated on the x axis. The numbers negative 25 through 100, in increments of 25, are indicated on the y axis. The curve begins to the left of the y axis, just above the x axis. As it moves rightward, it goes upward, at first so slowly it is almost horizontal, then more and more quickly. It ends to the right of the y axis, high above the x axis. The curve crosses the y axis a little above the origin, and it passes through the point with coordinates 1 comma 8, the point with coordinates 2 comma 16, and the point with coordinates 3 comma 32. It ends to the right of the y-axis, high above the x-axis.

     

Show Answer Correct Answer: D

Choice D is correct. The y-intercept of the graph of an equation is the point with coordinates 0 comma b, where b is the value of y when x equals 0. For the given equation, y equals 4 when x equals 0. It follows that the y-intercept of the graph of the given equation is the point with coordinates 0 comma 4. Additionally, for the given equation, the value of y doubles for each increase of 1 in the value of x. Therefore, the graph contains the points with coordinates 1 comma 8, 2 comma 16, 3 comma 32, and 4 comma 64. Only the graph shown in choice D passes through these points.

Choices A and B are incorrect because these are graphs of decreasing, not increasing, exponential functions. Choice C is incorrect because the value of y increases by a growth factor greater than 2 for each increase of 1 in the value of x.

 

Question 208 208 of 479 selected Nonlinear Functions E

  • Moving from left to right:
    • The curve passes from quadrant 2 to quadrant 1 to quadrant 4.
    • In quadrant 2, the curve trends down gradually to the approximate point (0 comma 2.5).
    • In quadrant 1, the curve trends down sharply to point (5 comma 0).
    • In quadrant 4, the curve trends down sharply.
  • The curve passes through the following points:
    • approximately (0 comma 2.5)
    • approximately (4 comma 0.8)

What is the x-intercept of the graph shown?

  1. (-5,0)

  2. (5,0)

  3. (-2,0)

  4. (2,0)

Show Answer Correct Answer: B

Choice B is correct. An x-intercept of a graph in the xy-plane is a point at which the graph crosses the x-axis. The graph shown crosses the x-axis at the point (5,0). Therefore, the x-intercept of the graph shown is (5,0).

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 209 209 of 479 selected Equivalent Expressions H

The expression one-third x squared, minus 2 can be rewritten as one-third times, open parenthesis, x minus k, close parenthesis, times, open parenthesis, x plus k, close parenthesis, where k is a positive constant. What is the value of k ?

  1. 2

  2. 6

  3. the square root of 2

  4. the square root of 6

Show Answer Correct Answer: D

Choice D is correct. Factoring out the coefficient one third, the given expression can be rewritten as one third times, open parenthesis, x squared, minus 6, close parenthesis. The expression x squared, minus 6 can be approached as a difference of squares and rewritten as open parenthesis, x minus the square root of 6, close parenthesis, times, open parenthesis, x plus the square root of 6, close parenthesis. Therefore, k must be the square root of 6.

Choice A is incorrect. If k were 2, then the expression given would be rewritten as one third times, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 2, close parenthesis, which is equivalent to one third, x squared, minus four thirds, not one third, x squared, minus 2.

Choice B is incorrect. This may result from incorrectly factoring the expression and finding open parenthesis, x minus 6, close parenthesis, times, open parenthesis, x plus 6, close parenthesis as the factored form of the expression. Choice C is incorrect. This may result from incorrectly distributing the one third and rewriting the expression as one third times, open parenthesis, x squared, minus 2, close parenthesis.

 

Question 210 210 of 479 selected Nonlinear Functions E

If f of x equals, the fraction with numerator x squared minus 6 x plus 3, and denominator x minus 1, end fraction, what is f of negative 1 ?

  1. –5

  2. –2

  3. 2

  4. 5

Show Answer Correct Answer: A

Choice A is correct. Substituting –1 for x in the equation that defines f gives f of negative 1 equals, the fraction with numerator, open parenthesis, negative 1, close parenthesis, squared, minus 6, times, open parenthesis, negative 1, close parenthesis, plus 3, and denominator negative 1 minus 1, end fraction. Simplifying the expressions in the numerator and denominator yields the fraction with numerator 1 plus 6, plus 3, and denominator negative 2, which is equal to 10 over negative 2 or –5.

Choices B, C, and D are incorrect and may result from misapplying the order of operations when substituting –1 for x.

 

Question 211 211 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

5 x 2 + 10 x + 16 = 0

How many distinct real solutions does the given equation have?

  1. Exactly one

  2. Exactly two

  3. Infinitely many

  4. Zero

Show Answer Correct Answer: D

Choice D is correct. The number of solutions of a quadratic equation of the form ax2+bx+c=0, where a , b , and c are constants, can be determined by the value of the discriminant, b2-4ac. If the value of the discriminant is positive, then the quadratic equation has exactly two distinct real solutions. If the value of the discriminant is equal to zero, then the quadratic equation has exactly one real solution. If the value of the discriminant is negative, then the quadratic equation has zero real solutions. In the given equation, 5x2+10x+16=0, a= 5, b=10, and c=16. Substituting these values for a , b , and c in b2-4ac yields (10)2-4(5)(16), or -220. Since the value of its discriminant is negative, the given equation has zero real solutions. Therefore, the number of distinct real solutions the given equation has is zero.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 212 212 of 479 selected Nonlinear Functions H

The function f(x)=19(x-7)2+3 gives a metal ball's height above the ground f(x), in inches, x seconds after it started moving on a track, where 0x10. Which of the following is the best interpretation of the vertex of the graph of y=f(x) in the xy-plane?

  1. The metal ball's minimum height was 3 inches above the ground.

  2. The metal ball's minimum height was 7 inches above the ground.

  3. The metal ball's height was 3 inches above the ground when it started moving.

  4. The metal ball's height was 7 inches above the ground when it started moving.

Show Answer Correct Answer: A

Choice A is correct. The graph of a quadratic equation in the form y=a(x-h)2+k, where a , h , and k are positive constants, is a parabola that opens upward with vertex (h,k). The given function f(x)=19(x-7)2+3 is in the form y=a(x-h)2+k, where y=f(x), a=19, h = 7 , and k = 3 . Therefore, the graph of y=f(x) is a parabola that opens upward with vertex (7,3). Since the parabola opens upward, the vertex is the lowest point on the graph. It follows that the y-coordinate of the vertex of the graph of y=f(x) is the minimum value of f(x). Therefore, the minimum value of f(x) is 3 . It’s given that f(x)=19(x-7)2+3 represents the metal ball’s height above the ground, in inches, x seconds after it started moving on a track. Therefore, the best interpretation of the vertex of the graph of y=f(x) is that the metal ball’s minimum height was 3 inches above the ground.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 213 213 of 479 selected Nonlinear Functions M

The area of a triangle is 270 square centimeters. The length of the base of the triangle is 12 centimeters greater than the height of the triangle. What is the height, in centimeters, of the triangle?

  1. 15

  2. 18

  3. 30

  4. 36

Show Answer Correct Answer: B

Choice B is correct. The area, A, of a triangle is given by the formula A=12bh, where b represents the length of the base of the triangle and h represents its height. It’s given that the area of a triangle is 270 square centimeters and that the length of the base of this triangle is 12 centimeters greater than the height of the triangle. Let x represent the height, in centimeters, of the triangle. It follows that the length of the base of the triangle can be expressed as x+12. Substituting 270 for A, x for h, and x+12 for b in the formula A=12bh yields 270=12(x+12)(x), or 270=12x(x+12). Multiplying both sides of this equation by 2 yields 540=x(x+12). Applying the distributive property on the right-hand side of this equation yields 540=x2+12x. Subtracting 540 from both sides of this equation yields 0=x2+12x-540. In factored form, this equation is equivalent to (x+30)(x-18)=0. Applying the zero product property, it follows that x+30=0 or x-18=0. Subtracting 30 from both sides of the equation x+30=0 yields x=-30. Adding 18 to both sides of the equation x-18=0 yields x=18. Since x represents the height of the triangle, it must be positive. Therefore, the height, in centimeters, of the triangle is 18.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 214 214 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

y equals, x squared plus 2 x plus 1, and, x plus y plus 1, equals 0

If the ordered pair x subscript 1 comma y subscript 1 and the ordered pair x subscript 2 comma y subscript 2 are the two solutions to the system of equations above, what is the value of y subscript 1, plus, y subscript 2 ?

  1. negative 3
  2. negative 2
  3. negative 1
  4. 1
Show Answer Correct Answer: D

Choice D is correct. The system of equations can be solved using the substitution method. Solving the second equation for y gives y = –x – 1. Substituting the expression –x – 1 for y into the first equation gives –x – 1 = x2 + 2x + 1. Adding x + 1 to both sides of the equation yields x2 + 3x + 2 = 0. The left-hand side of the equation can be factored by finding two numbers whose sum is 3 and whose product is 2, which gives (x + 2)(x + 1) = 0. Setting each factor equal to 0 yields x + 2 = 0 and x + 1 = 0, and solving for x yields x = –2 or x = –1. These values of x can be substituted for x in the equation y = –x – 1 to find the corresponding y-values: y = –(–2) – 1 = 2 – 1 = 1 and y = –(–1) – 1 = 1 – 1 = 0. It follows that (–2, 1) and (–1, 0) are the solutions to the given system of equations. Therefore, (x1, y1) = (–2, 1), (x2, y2) = (–1, 0), and y1 + y2 = 1 + 0 = 1.

Choice A is incorrect. The solutions to the system of equations are (x1, y1) = (–2, 1) and (x2, y2) = (–1, 0). Therefore, –3 is the sum of the x-coordinates of the solutions, not the sum of the y-coordinates of the solutions. Choices B and C are incorrect and may be the result of computation or substitution errors.

Question 215 215 of 479 selected Nonlinear Functions M

The function f(w)=6w2 gives the area of a rectangle, in square feet (ft2), if its width is w ft and its length is 6 times its width. Which of the following is the best interpretation of f(14)=1,176 ?

  1. If the width of the rectangle is 14 ft, then the area of the rectangle is 1,176 ft2.

  2. If the width of the rectangle is 14 ft, then the length of the rectangle is 1,176 ft.

  3. If the width of the rectangle is 1,176 ft, then the length of the rectangle is 14 ft.

  4. If the width of the rectangle is 1,176 ft, then the area of the rectangle is 14 ft2.

Show Answer Correct Answer: A

Choice A is correct. The function f gives the area of the rectangle, in ft2, if its width is w ft. Since the value of f(14) is the value of f(w) if w = 14 , it follows that f(14)=1,176 means that f(w) is 1,176 if w = 14 . In the given context, this means that if the width of the rectangle is 14 ft, then the area of the rectangle is 1,176 ft2.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from interpreting f(w) as the width, in ft, of the rectangle if its area is w ft2, rather than as the area, in ft2, of the rectangle if its width is w ft.

Question 216 216 of 479 selected Equivalent Expressions E

The expression 2 x squared, plus a, x is equivalent to x times, open parenthesis, 2 x, plus 7, close parenthesis  for some constant a. What is the value of a ?

  1. 2

  2. 3

  3. 4

  4. 7

Show Answer Correct Answer: D

Choice D is correct. It’s given that 2 x squared, plus a, x is equivalent to x times, open parenthesis, 2 x plus 7, close parenthesis for some constant a. Distributing the x over each term in the parentheses gives 2 x squared, plus 7 x, which is in the same form as the first given expression, 2 x squared, plus a, x. The coefficient of the second term in 2 x squared, plus 7 x is 7. Therefore, the value of a is 7.

Choice A is incorrect. If the value of a were 2, then 2 x squared, plus a, x would be equivalent to 2 x squared, plus 2 x, which isn’t equivalent to x times, open parenthesis, 2 x plus 7, close parenthesis. Choice B is incorrect. If the value of a were 3, then 2 x squared, plus a, x would be equivalent to 2 x squared, plus 3 x, which isn’t equivalent to x times, open parenthesis, 2 x plus 7, close parenthesis. Choice C is incorrect. If the value of a were 4, then 2 x squared, plus a, x would be equivalent to 2 x squared, plus 4 x, which isn’t equivalent to x times, open parenthesis, 2 x plus 7, close parenthesis.

 

Question 217 217 of 479 selected Nonlinear Functions M
x f(x)
-1 10
0 14
1 20

For the quadratic function f , the table shows three values of x and their corresponding values of f(x). Which equation defines f ?

  1. f(x)=3x2+3x+14

  2. f(x)=5x2+x+14

  3. f(x)=9x2-x+14

  4. f(x)=x2+5x+14

Show Answer Correct Answer: D

Choice D is correct. The equation of a quadratic function can be written in the form f(x)=a(x-h)2+k, where a , h , and k are constants. It’s given in the table that when x = -1 , the corresponding value of f(x) is 10 . Substituting -1 for x and 10 for f(x) in the equation f(x)=a(x-h)2+k gives 10=a(-1-h)2+k, which is equivalent to 10=a(1+2h+h2)+k, or 10=a+2ah+ah2+k. It’s given in the table that when x = 0 , the corresponding value of f(x) is 14 . Substituting 0 for x and 14 for f(x) in the equation f(x)=a(x-h)2+k gives 14=a(0-h)2+k, or 14=ah2+k. It’s given in the table that when x = 1 , the corresponding value of f(x) is 20 . Substituting 1 for x and 20 for f(x) in the equation f(x)=a(x-h)2+k gives 20=a(1-h)2+k, which is equivalent to 20=a(1-2h+h2)+k, or 20=a-2ah+ah2+k. Adding 20=a-2ah+ah2+k to the equation 10=a+2ah+ah2+k gives 30=2a+2ah2+2k. Dividing both sides of this equation by 2 gives 15=a+ah2+k. Since 14=ah2+k, substituting 14 for ah2+k  into the equation 15=a+ah2+k gives 15=a+14. Subtracting 14 from both sides of this equation gives a = 1 . Substituting 1 for a in the equations 14 = a h 2 + k and 20 = a h 2 - 2 a h + a + k gives 14=h2+k and 20=1-2h+h2+k, respectively. Since 14 = h 2 + k , substituting 14 for h 2 + k in the equation 20=1-2h+h2+k gives 20=1-2h+14, or 20=15-2h. Subtracting 15 from both sides of this equation gives 5=-2h. Dividing both sides of this equation by -2 gives -52=h. Substituting - 5 2 for h into the equation 14=h2+k gives 14=(-52)2+k, or 14=254+k. Subtracting 254 from both sides of this equation gives 314=k. Substituting 1 for a -52 for h , and 314 for k in the equation f(x)=a(x-h)2+k gives f(x)=(x+52)2+314, which is equivalent to f(x)=x2+5x+254+314, or f(x)=x2+5x+14. Therefore, f(x)=x2+5x+14 defines f .

Choice A is incorrect. If f(x)=3x2+3x+14, then when x = -1 , the corresponding value of f(x) is 14 , not 10 .

Choice B is incorrect. If f(x)=5x2+x+14, then when x = -1 , the corresponding value of f(x) is 18 , not 10 .

Choice C is incorrect. If f(x)=9x2-x+14, then when x = -1 , the corresponding value of f(x) is 24 , not 10 , and when x = 1 , the corresponding value of f(x) is 22 , not 20 .

Question 218 218 of 479 selected Nonlinear Functions H

f(t)=8,000(0.65)t

The given function f models the number of coupons a company sent to their customers at the end of each year, where t represents the number of years since the end of 1998, and 0t5. If y=f(t) is graphed in the ty-plane, which of the following is the best interpretation of the y-intercept of the graph in this context?

  1. The minimum estimated number of coupons the company sent to their customers during the 5 years was 1,428 .

  2. The minimum estimated number of coupons the company sent to their customers during the 5 years was 8,000 .

  3. The estimated number of coupons the company sent to their customers at the end of 1998 was 1,428 .

  4. The estimated number of coupons the company sent to their customers at the end of 1998 was 8,000 .

Show Answer Correct Answer: D

Choice D is correct. The y-intercept of a graph in the ty-plane is the point where t = 0 . For the given function f, the y-intercept of the graph of y=f(t) in the ty-plane can be found by substituting 0 for t in the equation y=8,000(0.65)t, which gives y=8,000(0.65)0. This is equivalent to y=8,000(1), or y = 8,000 . Therefore, the y-intercept of the graph of y=f(t) is (0,8,000). It’s given that the function f models the number of coupons a company sent to their customers at the end of each year. Therefore, f(t) represents the estimated number of coupons the company sent to their customers at the end of each year. It's also given that t represents the number of years since the end of 1998. Therefore, t = 0 represents 0 years since the end of 1998, or the end of 1998. Thus, the best interpretation of the y-intercept of the graph of y=f(t) is that the estimated number of coupons the company sent to their customers at the end of 1998 was 8,000 .

Choice A is incorrect and may result from conceptual or calculation errors. 

Choice B is incorrect and may result from conceptual or calculation errors. 

Choice C is incorrect and may result from conceptual or calculation errors. 

Question 219 219 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

  • For the curve in the system:
    • The curve passes from quadrant 2 to quadrant 1.
    • In quadrant 2, the curve trends up gradually to the approximate point (0 comma 4.063).
    • In quadrant 1:
      • The curve trends up gradually to point (4 comma 5).
      • The curve then trends up sharply.
    • The curve passes through the following points:
      • (3 comma nine halves)
      • (4 comma 5)
      • (5 comma 6)
  • For the line in the system:
    • The line slants sharply down from left to right.
    • The line passes through the following points:
      • (3 comma 9)
      • (4 comma 5)

The graph of a system of a linear equation and a nonlinear equation is shown. What is the solution (x,y) to this system?

  1. (0,0)

  2. (0,4)

  3. (4,5)

  4. (5,0)

Show Answer Correct Answer: C

Choice C is correct. The solution to the system of two equations corresponds to the point where the graphs of the equations intersect. The graphs of the linear equation and the nonlinear equation shown intersect at the point (4,5). Thus, the solution to the system is (4,5).

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 220 220 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

If u minus 3 equals, the fraction 6 over, t minus 2, end fraction , what is t in terms of u ?

  1. t equals, the fraction 1 over u

  2. t equals, the fraction 2 u plus 9, over u

  3. t equals, the fraction 1 over, u minus 3, end fraction

  4. t equals, the fraction 2 u, over u minus 3, end fraction

Show Answer Correct Answer: D

Choice D is correct. Multiplying both sides of the given equation by t minus 2 yields open parenthesis, t minus 2, close parenthesis, times, open parenthesis, u minus 3, close parenthesis, equals 6. Dividing both sides of this equation by u minus 3 yields t minus 2, equals, the fraction with numerator 6, and denominator u minus 3, end fraction. Adding 2 to both sides of this equation yields t equals, the fraction with numerator 6, and denominator u minus 3, end fraction, plus 2, which can be rewritten as t equals, the fraction with numerator 6, and denominator u minus 3, end fraction, plus, the fraction with numerator 2 times, open parenthesis, u minus 3, close parenthesis, and denominator u minus 3, end fraction. Since the fractions on the right-hand side of this equation have a common denominator, adding the fractions yields t equals, the fraction with numerator 6 plus 2, times, open parenthesis, u minus 3, close parenthesis, and denominator u minus 3, end fraction. Applying the distributive property to the numerator on the right-hand side of this equation yields t equals, the fraction with numerator 6 plus 2 u, minus 6, and denominator u minus 3, end fraction, which is equivalent to t equals, the fraction with numerator 2 u, and denominator u minus 3, end fraction.

Choices A, B, and C are incorrect and may result from various misconceptions or miscalculations.

 

Question 221 221 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

- 9 x 2 + 30 x + c = 0

In the given equation, c is a constant. The equation has exactly one solution. What is the value of c ?

  1. 3

  2. 0

  3. -25

  4. -53

Show Answer Correct Answer: C

Choice C is correct. It's given that the equation -9x2+30x+c=0 has exactly one solution. A quadratic equation of the form a x 2 + b x + c = 0 has exactly one solution if and only if its discriminant, - 4 a c + b 2 , is equal to zero. It follows that for the given equation, a = -9 and b = 30 . Substituting -9 for a and 30 for b into b2-4ac  yields 302-4(-9)(c), or 900+36c. Since the discriminant must equal zero, 900+36c=0. Subtracting 36 c from both sides of this equation yields 900 = - 36 c . Dividing each side of this equation by -36 yields -25 = c . Therefore, the value of c is -25

Choice A is incorrect. If the value of c is 3 , this would yield a discriminant that is greater than zero. Therefore, the given equation would have two solutions, rather than exactly one solution.

Choice B is incorrect. If the value of c is 0 , this would yield a discriminant that is greater than zero. Therefore, the given equation would have two solutions, rather than exactly one solution.

Choice D is incorrect. If the value of c is -53 , this would yield a discriminant that is less than zero. Therefore, the given equation would have no real solutions, rather than exactly one solution.

Question 222 222 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

x 2 x 2 - c 2 = c 2 x 2 - c 2 + 39

In the given equation, c is a positive constant. Which of the following is one of the solutions to the given equation?

  1. - c

  2. -c2-392

  3. -392-c2

  4. -c2+392

Show Answer Correct Answer: D

Choice D is correct. If x2-c20, then neither side of the given equation is defined and there can be no solution. Therefore, x2-c2>0. Subtracting c2x2-c2 from both sides of the given equation yields x2x2-c2-c2x2-c2=39, or x2-c2x2-c2=39. Squaring both sides of this equation yields (x2-c2x2-c2)2=392, or (x2-c2)(x2-c2)x2-c2=392. Since x2-c2 is positive and, therefore, nonzero, the expression x2-c2x2-c2 is defined and equivalent to 1 . It follows that the equation (x2-c2)(x2-c2)x2-c2=392 can be rewritten as (x2-c2x2-c2)(x2-c2)=392, or (1)(x2-c2)=392, which is equivalent to x2-c2=392. Adding c2 to both sides of this equation yields x2=c2+392. Taking the square root of both sides of this equation yields two solutions: x=c2+392 and x=-c2+392. Therefore, of the given choices, -c2+392 is one of the solutions to the given equation.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 223 223 of 479 selected Nonlinear Functions H

Which of the following functions has(have) a minimum value at -3 ?

  1. f(x)=-6(3)x-3
  2. g(x)=-3(6)x

 

  1. I only

  2. II only

  3. I and II

  4. Neither I nor II

Show Answer Correct Answer: D

Choice D is correct. A function of the form f(x)=a(b)x+c, where a<0 and b>1, is a decreasing function. Both of the given functions are of this form; therefore, both are decreasing functions. If a function f is decreasing as the value of x increases, the corresponding value of f(x) decreases; therefore, the function doesn’t have a minimum value. Thus, neither of the given functions has a minimum value.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 224 224 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

3x(x-4)(x+5)=0

What is one of the solutions to the given equation?

  1. -4

  2. 0

  3. 3

  4. 5

Show Answer Correct Answer: B

Choice B is correct. Applying the zero product property to the given equation yields 3 x = 0 , x - 4 = 0 , and x + 5 = 0 . Dividing each side of the equation 3 x = 0 by 3 yields x = 0 . Adding 4 to each side of the equation x - 4 = 0 yields x = 4 . Subtracting 5 from each side of the equation x + 5 = 0 yields x = - 5 . Therefore, the solutions to the given equation are 0 , 4 , and - 5 . Thus, one of the solutions to the given equation is 0 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 225 225 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

w 2 + 12 w - 40 = 0

Which of the following is a solution to the given equation?

  1. 6 - 2 19

  2. 2 19

  3. 19

  4. -6 + 2 19

Show Answer Correct Answer: D

Choice D is correct. Adding 40 to both sides of the given equation yields w 2 + 12 w = 40 . To complete the square, adding (122)2, or 62, to both sides of this equation yields w2+12w+62=40+62, or (w+6)2=76. Taking the square root of both sides of this equation yields w+6=±76, or w+6=±219. Subtracting 6 from both sides of this equation yields w=-6±219. Therefore, the solutions to the given equation are -6+219 and -6-219. Of these two solutions, only -6+219 is given as a choice.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 226 226 of 479 selected Nonlinear Functions E

f of x equals, 2 times, open parenthesis, 3 raised to the x power, close parenthesis

For the function f defined above, what is the value of f of 2?

  1. 9

  2. 12

  3. 18

  4. 36

Show Answer Correct Answer: C

Choice C is correct. The value of f of 2 is found by evaluating the expression 2 times 3 raised to the x power when x equals 2. Substituting 2 for x in the given equation yields f of 2 equals, 2 times, 3 squared. Simplifying 3 squared in the equation results in f of 2 equals, 2 times 9. Evaluating the right-hand side of the equation yields f of 2 equals 18. Therefore, the value of f of 2 is 18.

Choice A is incorrect and may result from evaluating the expression as 3 squared. Choice B is incorrect and may result from evaluating the expression as 2 times, open parenthesis, 3 times 2, close parenthesis. Choice D is incorrect and may result from evaluating the expression as open parenthesis, 2 times 3, close parenthesis, squared.

 

Question 227 227 of 479 selected Nonlinear Functions M

An object’s kinetic energy, in joules, is equal to the product of one-half the object’s mass, in kilograms, and the square of the object’s speed, in meters per second. What is the speed, in meters per second, of an object with a mass of 4 kilograms and kinetic energy of 18 joules?

  1. 3

  2. 6

  3. 9

  4. 36

Show Answer Correct Answer: A

Choice A is correct. It’s given that an object’s kinetic energy, in joules, is equal to the product of one-half the object’s mass, in kilograms, and the square of the object’s speed, in meters per second. This relationship can be represented by the equation K equals, one half m v squared, where K is the kinetic energy, m is the mass, and v is the speed. Substituting a mass of 4 kilograms for m and a kinetic energy of 18 joules for K results in the equation 18 equals, one half times 4, times v squared, or 18 equals, 2 v squared. Dividing both sides of this equation by 2 yields 9 equals, v squared. Taking the square root of both sides yields v equals, negative 3 and v equals 3. Since speed can’t be expressed as a negative number, the speed of the object is 3 meters per second.

Choice B is incorrect and may result from computation errors. Choice C is incorrect. This is the value of v squared  rather than v. Choice D is incorrect. This is the value of 4 v squared  rather than v.

 

Question 228 228 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

  • For the absolute value function:
    • Moving from left to right:
      • The function slants sharply down to the point (4 comma 2).
      • The function then slants sharply up.
    • The function passes through the following points:
      • (1 comma 5) 
      • (4 comma 2)
      • (6 comma 4)
  • For the linear function:
    • The function slants sharply up from left to right.
    • The function passes through the following points:
      • (0 comma 4)
      • (1 comma 5)

The graph of a system of an absolute value function and a linear function is shown. What is the solution (x,y) to this system of two equations?

  1. (-1,5)

  2. (0,4)

  3. (1,5)

  4. (4,2)

Show Answer Correct Answer: C

Choice C is correct. The solution to the system of two equations corresponds to the point where the graphs of the equations intersect. The graphs of the linear function and the absolute value function shown intersect at the point (1,5). Thus, the solution to the system is (1,5).

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the y-intercept of the graph of the linear function.

Choice D is incorrect. This is the vertex of the graph of the absolute value function.

Question 229 229 of 479 selected Nonlinear Functions H

In the xy-plane, a parabola has vertex (9,-14) and intersects the x-axis at two points. If the equation of the parabola is written in the form y = a x 2 + b x + c , where a , b , and c are constants, which of the following could be the value of a + b + c ?

  1. -23

  2. -19

  3. -14

  4. -12

Show Answer Correct Answer: D

Choice D is correct. The equation of a parabola in the xy-plane can be written in the form y=a(x-h)2+k, where a is a constant and (h,k) is the vertex of the parabola. If a is positive, the parabola will open upward, and if a is negative, the parabola will open downward. It’s given that the parabola has vertex (9,-14). Substituting 9 for h and -14 for k in the equation y=a(x-h)2+k gives y=a(x-9)2-14, which can be rewritten as y=a(x-9)(x-9)-14, or y=a(x2-18x+81)-14. Distributing the factor of a on the right-hand side of this equation yields y=ax2-18ax+81a-14. Therefore, the equation of the parabola, y=ax2-18ax+81a-14, can be written in the form y=ax2+bx+c, where a=a, b=-18a, and c=81a-14. Substituting - 18 a for b and 81 a - 14 for c in the expression a+b+c yields (a)+(-18a)+(81a-14), or 64a-14. Since the vertex of the parabola, (9,-14), is below the x-axis, and it’s given that the parabola intersects the x-axis at two points, the parabola must open upward. Therefore, the constant a must have a positive value. Setting the expression 64 a - 14 equal to the value in choice D yields 64a-14=-12. Adding 14 to both sides of this equation yields 64 a = 2 . Dividing both sides of this equation by 64 yields a=264, which is a positive value. Therefore, if the equation of the parabola is written in the form y=ax2+bx+c, where a , b , and c are constants, the value of a+b+c could be -12.

Choice A is incorrect. If the equation of a parabola with a vertex at (9,-14) is written in the form y=ax2+bx+c, where a , b , and c are constants and a+b+c=-23, then the value of a will be negative, which means the parabola will open downward, not upward, and will intersect the x-axis at zero points, not two points. 

Choice B is incorrect. If the equation of a parabola with a vertex at (9,-14) is written in the form y=ax2+bx+c, where a , b , and c are constants and a+b+c=-19, then the value of a will be negative, which means the parabola will open downward, not upward, and will intersect the x-axis at zero points, not two points.

Choice C is incorrect. If the equation of a parabola with a vertex at (9,-14) is written in the form y=ax2+bx+c, where a , b , and c are constants and a+b+c=-14, then the value of a will be 0 , which is inconsistent with the equation of a parabola.

Question 230 230 of 479 selected Nonlinear Functions E

  • Moving from left to right:
    • The curve passes from quadrant 2 to quadrant 1.
    • In quadrant 2, the curve trends up gradually to point (0 comma 8).
    • In quadrant 1, the curve trends up sharply.
  • The curve passes through the following points:
    • (0 comma 8)
    • (1 comma 9)

What is the y-intercept of the graph shown?

  1. (-8,0)

  2. (-6,0)

  3. (0,6)

  4. (0,8)

Show Answer Correct Answer: D

Choice D is correct. The y-intercept of a graph in the xy-plane is the point at which the graph crosses the y-axis. The graph shown crosses the y-axis at the point (0,8). Therefore, the y-intercept of the graph shown is (0,8).

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 231 231 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

8j=k+15m

The given equation relates the distinct positive numbers j , k , and m . Which equation correctly expresses j in terms of k and m ?

  1. j=k8+15m

  2. j=k+15m8

  3. j=8(k+15m)

  4. j=k+15m8

Show Answer Correct Answer: D

Choice D is correct. To express j in terms of k and m , the given equation must be solved for j . Dividing each side of the given equation by 8 yields  j=k+15m8.

Choice A is incorrect. This is equivalent to 8j=k+120m.

Choice B is incorrect. This is equivalent to 8j=8k+15m.

Choice C is incorrect. This is equivalent to j8=k+15m.

Question 232 232 of 479 selected Nonlinear Functions E

  • The parabola opens upward.
  • The vertex is at the point (0 comma 7).
  • The parabola passes through the following points:
    • (negative 1 comma StartFraction 26 Over 3 EndFraction)
    • (0 comma 7)
    • (1 comma StartFraction 26 Over 3 EndFraction)

The parabola shown intersects the y-axis at the point (x,y). What is the value of y ?

Show Answer Correct Answer: 7

The correct answer is 7 . It's given that the parabola intersects the y-axis at the point (x,y). The graph shows that the parabola intersects the y-axis at the point (0,7). Therefore, the value of y is 7 .

Question 233 233 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

8 x + y = -11

2 x 2 = y + 341

The graphs of the equations in the given system of equations intersect at the point (x , y ) in the x y -plane. What is a possible value of x ?

  1. -15

  2. -11

  3. 2

  4. 8

Show Answer Correct Answer: A

Choice A is correct. It's given that the graphs of the equations in the given system of equations intersect at the point (x,y). Therefore, this intersection point is a solution to the given system. The solution can be found by isolating y in each equation. The given equation 8x+y=-11 can be rewritten to isolate y by subtracting 8 x from both sides of the equation, which gives y=-8x-11. The given equation 2x2=y+341 can be rewritten to isolate y by subtracting 341 from both sides of the equation, which gives 2x2-341=y. With each equation solved for y , the value of y from one equation can be substituted into the other, which gives 2x2-341=-8x-11. Adding 8 x and 11 to both sides of this equation results in 2x2+8x-330=0. Dividing both sides of this equation by 2 results in x2+4x-165=0. This equation can be rewritten by factoring the left-hand side, which yields (x+15)(x-11)=0. By the zero-product property, if (x+15)(x-11)=0, then (x+15)=0, or (x-11)=0. It follows that x=-15, or x=11. Since only -15 is given as a choice, a possible value of x is -15

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 234 234 of 479 selected Nonlinear Functions H
xp(x)
x is negative 2p of x is 5
x is negative 1p of x is 0
x is 0p of x is negative 3
x is 1p of x is negative 1
x is 2p of x is 0

The table above gives selected values of a polynomial function p. Based on the values in the table, which of the following must be a factor of p ?

  1. open parenthesis, x minus 3, close parenthesis

  2. open parenthesis, x plus 3, close parenthesis

  3. open parenthesis, x minus 1, close parenthesis, times, open parenthesis, x plus 2, close parenthesis

  4. open parenthesis, x plus 1, close parenthesis, times, open parenthesis, x minus 2, close parenthesis

Show Answer Correct Answer: D

Choice D is correct. According to the table, when x is negative 1 or 2, p of x, equals 0. Therefore, two x-intercepts of the graph of p are the points with coordinates negative 1 comma 0 and 2 comma 0. Since the points with coordinates negative 1 comma 0 and 2 comma 0 are x-intercepts, it follows that open parenthesis, x plus 1, close parenthesis and open parenthesis, x minus 2, close parenthesis are factors of the polynomial equation. This is because when x equals negative 1, the value of x plus 1 is 0. Similarly, when x equals 2, the value of x minus 2 is 0. Therefore, the product open parenthesis, x plus 1, close parenthesis, times, open parenthesis, x minus 2, close parenthesis is a factor of the polynomial function p.

Choice A is incorrect. The factor x minus 3 corresponds to an x-intercept of the point with coordinates 3 comma 0, which isn’t present in the table. Choice B is incorrect. The factor x plus 3 corresponds to an x-intercept of the point with coordinates negative 3 comma 0, which isn’t present in the table. Choice C is incorrect. The factors x minus 1 and x plus 2 correspond to x-intercepts with coordinates 1 comma 0 and negative 2 comma 0, respectively, which aren’t present in the table.

 

Question 235 235 of 479 selected Nonlinear Functions E

A ball is dropped from an initial height of 22 feet and bounces off the ground repeatedly. The function h estimates that the maximum height reached after each time the ball hits the ground is 85 % of the maximum height reached after the previous time the ball hit the ground. Which equation defines h , where h(n) is the estimated maximum height of the ball after it has hit the ground n times and n is a whole number greater than 1 and less than 10 ?

  1. h(n)=22(0.22)n

  2. h(n)=22(0.85)n

  3. h(n)=85(0.22)n

  4. h(n)=85(0.85)n

Show Answer Correct Answer: B

Choice B is correct. It's given that for the function h , h(n) is the estimated maximum height, in feet, of the ball after it has hit the ground n times. It's also given that the function h estimates that the maximum height reached after each time the ball hits the ground is 85% of the maximum height reached after the previous time the ball hit the ground. It follows that h is a decreasing exponential function that can be written in the form h(n)=a(p100)n, where a is the initial height, in feet, the ball was dropped from and the function estimates that the maximum height reached after each time the ball hits the ground is p% of the maximum height reached after the previous time the ball hit the ground. It's given that the ball is dropped from an initial height of 22 feet. Therefore, a = 22 . Since the function h estimates that the maximum height reached after each time the ball hits the ground is 85% of the maximum height reached after the previous time the ball hit the ground, p = 85 . Substituting 22 for a and 85 for p in the equation h(n)=a(p100)n yields h(n)=22(85100)n, or h(n)=22(0.85)n.

Choice A is incorrect. This function estimates that the maximum height reached after each time the ball hits the ground is 22%, not 85%, of the maximum height reached after the previous time the ball hit the ground.

Choice C is incorrect. This function estimates that the ball is dropped from an initial height of 85 feet, not 22 feet, and that the maximum height reached after each time the ball hits the ground is 22%, not 85%, of the maximum height reached after the previous time the ball hit the ground.

Choice D is incorrect. This function estimates that the ball is dropped from an initial height of 85 feet, not 22 feet.

Question 236 236 of 479 selected Nonlinear Functions E

  • Moving from left to right:
    • The curve passes from quadrant 3 to quadrant 4.
    • In quadrant 3, the curve trends up sharply to point (0 comma negative 6).
    • In quadrant 4, the curve trends up gradually.
  • As x increases, the curve approaches the line y equals negative 5.
  • The curve passes through the following points:
    • (0 comma negative 6)
    • (1 comma negative 5 and one sixth)

What is the y-intercept of the graph shown?

  1. (0,-6)

  2. (-6,0)

  3. (0,0)

  4. (-5,-5)

Show Answer Correct Answer: A

Choice A is correct. The y-intercept of a graph in the xy-plane is the point (x,y) on the graph where x = 0 . For the graph shown, at x = 0 , the corresponding value of y is -6 . Therefore, the y-intercept of the graph shown is (0,-6).

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 237 237 of 479 selected Nonlinear Functions M

The height, in feet, of an object x seconds after it is thrown straight up in the air can be modeled by the function h of x equals, negative 16 x squared, plus 20 x, plus 5. Based on the model, which of the following statements best interprets the equation h of 1 point 4 equals, 1 point 6 4 ?

  1. The height of the object 1.4 seconds after being thrown straight up in the air is 1.64 feet.

  2. The height of the object 1.64 seconds after being thrown straight up in the air is 1.4 feet.

  3. The height of the object 1.64 seconds after being thrown straight up in the air is approximately 1.4 times as great as its initial height.

  4. The speed of the object 1.4 seconds after being thrown straight up in the air is approximately 1.64 feet per second.

Show Answer Correct Answer: A

Choice A is correct. The value 1.4 is the value of x, which represents the number of seconds after the object was thrown straight up in the air. When the function h is evaluated for x = 1.4, the function has a value of 1.64, which is the height, in feet, of the object.

Choices B and C are incorrect and may result from misidentifying seconds as feet and feet as seconds. Additionally, choice C may result from incorrectly including the initial height of the object as the input x. Choice D is incorrect and may result from misidentifying height as speed.

Question 238 238 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

The speed of sound in dry air, v, can be modeled by the formula v equals, 331 point 3 plus 0 point 6 0 6 times T, where T is the temperature in degrees Celsius and v is measured in meters per second. Which of the following correctly expresses T in terms of v ?

  1. T equals, the fraction with numerator, v plus 0 point 6 0 6, and denominator, 331 point 3
  2. T equals, the fraction with numerator, v minus 0 point 6 0 6, and denominator, 331 point 3
  3. T equals, the fraction with numerator, v plus 331 point 3, and denominator, 0 point 6 0 6
  4. T equals, the fraction with numerator, v minus 331 point 3, and denominator, 0 point 6 0 6
Show Answer Correct Answer: D

Choice D is correct. To express T in terms of v, subtract 331.3 from both sides of the equation, which gives v – 331.3 = 0.606T. Dividing both sides of the equation by 0.606 gives the fraction with numerator v minus 331 point 3, and denominator 0 point 6 0 6, equals T.

Choices A, B, and C are incorrect and are the result of incorrect steps while solving for T.

Question 239 239 of 479 selected Nonlinear Functions E

The function f is defined by f(x)=x3+9. What is the value of f(2)?

  1. 14

  2. 15

  3. 17

  4. 18

Show Answer Correct Answer: C

Choice C is correct. It's given that f(x)=x3+9. Substituting 2 for x in this equation yields f(2)=(2)3+9. This is equivalent to f(2)=8+9, or f(2)=17.

Choice A is incorrect. This is the value of 2+3+9, not 23+9.

Choice B is incorrect. This is the value of 2(3)+9, not 23+9.

Choice D is incorrect. This is the value of 32+9, not 23+9.

Question 240 240 of 479 selected Equivalent Expressions M

Which expression is equivalent to (7x3+7x)-(6x3-3x)?

  1. x 3 + 10 x

  2. - 13 x 3 + 10 x

  3. - 13 x 3 + 4 x

  4. x 3 + 4 x

Show Answer Correct Answer: A

Choice A is correct. Applying the distributive property, the given expression can be written as 7x3+7x-6x3+3x. Grouping like terms in this expression yields (7x3-6x3)+(7x+3x). Combining like terms in this expression yields x3+10x.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 241 241 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

x ( x + 1 ) - 56 = 4 x ( x - 7 )

What is the sum of the solutions to the given equation?

Show Answer Correct Answer: 29/3, 9.666, 9.667

The correct answer is 293. Applying the distributive property to the left-hand side of the given equation, x(x+1)-56, yields x2+x-56. Applying the distributive property to the right-hand side of the given equation, 4x(x-7), yields 4x2-28x. Thus, the equation becomes x2+x-56=4x2-28x. Combining like terms on the left- and right-hand sides of this equation yields 0=(4x2-x2)+(-28x-x)+56, or 3x2-29x+56=0. For a quadratic equation in the form ax2+bx+c=0, where a , b , and c are constants, the quadratic formula gives the solutions to the equation in the form x=(-b±b2-4ac)2a. Substituting 3 for a , -29 for b , and 56 for c from the equation 3x2-29x+56=0 into the quadratic formula yields x=(29±(-29)2-4(3)(56))2(3), or x=296±136. It follows that the solutions to the given equation are 296+136 and 296-136. Adding these two solutions gives the sum of the solutions: 296+136+296-136, which is equivalent to 296+296, or 293. Note that 29/3, 9.666, and 9.667 are examples of ways to enter a correct answer.

Question 242 242 of 479 selected Nonlinear Functions E

During the first part of an experiment, a ball was launched from a 7 -foot-tall platform. The graph shows the height y , in feet, of the ball x seconds after it was launched during the first part of the experiment.

  • The parabola opens downward.
  • The vertex is at the approximate point (0.3 comma 8.3).
  • The parabola passes through the following points:
    • (0 comma 7)
    • approximately (0.3 comma 8.3)
    • approximately (0.6 comma 7)

During the second part of the experiment, the ball was launched the same way, but from a platform that is 2 feet shorter than the first platform. Which of the following graphs could represent the height y , in feet, of the ball x seconds after it was launched during the second part of the experiment?

    • The parabola opens downward.
    • The vertex is at the approximate point (0.3 comma 5.0).
    • The parabola passes through the following points:
      • approximately (0 comma 3.7)
      • approximately (0.3 comma 5.0)
      • approximately (0.6 comma 3.7)

    • The parabola opens downward.
    • The vertex is at the approximate point (0.3 comma 6.3).
    • The parabola passes through the following points:
      • (0 comma 5)
      • approximately (0.3 comma 6.3)
      • approximately (0.6 comma 5)

    • The parabola opens downward.
    • The vertex is at the approximate point (0.3 comma 10.3).
    • The parabola passes through the following points:
      • (0 comma 9)
      • approximately (0.3 comma 10.3)
      • approximately (0.6 comma 9)

    • The parabola opens downward.
    • The vertex is at the approximate point (0.3 comma 15.3).
    • The parabola passes through the following points:
      • (0 comma 14)
      • approximately (0.3 comma 15.3)
      • approximately (0.6 comma 14)

Show Answer Correct Answer: B

Choice B is correct. It's given that y represents the height, in feet, of the ball x seconds after it was launched. It's also given that during the first part of an experiment, a ball was launched from a 7 -foot-tall platform. Therefore, the y-coordinate of the y-intercept of the given graph, 7 , represents the platform height, in feet. During the second part of the experiment, the platform the ball was launched from was 2 feet shorter than the platform in the first part of the experiment. It follows that the height of the platform in the second part of the experiment was 7-2 feet, or 5 feet. Therefore, the y-coordinate of the y-intercept of the graph representing the second part of the experiment must be 5 . Only choice B satisfies this condition.

Choice A is incorrect. This could represent the graph if the ball were launched from a platform that was about 3 feet shorter rather than 2 feet shorter.

Choice C is incorrect. This could represent the graph if the ball were launched from a platform that was 2 feet taller rather than 2 feet shorter.

Choice D is incorrect. This could represent the graph if the ball were launched from a platform that was twice as tall rather than 2 feet shorter.

Question 243 243 of 479 selected Nonlinear Functions H

When the quadratic function f is graphed in the xy-plane, where y=f(x), its vertex is (-3,6). One of the x-intercepts of this graph is (-174,0). What is the other x-intercept of the graph?

  1. (-294,0)

  2. (-74,0)

  3. (54,0)

  4. (174,0)

Show Answer Correct Answer: B

Choice B is correct. Since the line of symmetry for the graph of a quadratic function contains the vertex of the graph, the x-coordinate of the vertex of the graph of y=f(x) is the x-coordinate of the midpoint of its two x-intercepts. The midpoint of two points with x-coordinates x1 and x2 has x-coordinate xm, where xm=x1+x22. It′s given that the vertex is (-3,6) and one of the x-intercepts is (-174,0). Substituting -3 for xm and -174 for x1 in the equation xm=x1+x22 yields -3=-174+x22. Multiplying each side of this equation by 2 yields -6=-174+x2. Adding 174 to each side of this equation yields -74=x2. Therefore, the other x-intercept is (-74,0).  

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 244 244 of 479 selected Nonlinear Functions H

Function f is a quadratic function where f(-20)=0 and f(-4)=0. The graph of y=f(x) in the xy-plane has a vertex at (r,-64). What is the value of r ?

Show Answer Correct Answer: -12

The correct answer is -12. It’s given that function f is a quadratic function where f(-20)=0 and f(-4)=0. It follows that the graph of y=f(x) in the xy-plane passes through the points (-20,0) and (-4,0). When the graph of a quadratic function contains two points (a,0) and (b,0), the x-coordinate of the vertex of the graph is the average of a and b. Therefore, the x-coordinate of the vertex of the graph of y=f(x) is -20+(-4)2, or -12. It's given that the graph of y=f(x) in the xy-plane has a vertex at (r,-64). It follows that the value of r is -12.

Question 245 245 of 479 selected Nonlinear Functions H

The function f is defined by f of x equals, open parenthesis, x plus 3, close parenthesis, times, open parenthesis, x plus 1, close parenthesis. The graph of f in the xy-plane is a parabola. Which of the following intervals contains the x-coordinate of the vertex of the graph of f ?

  1. negative 4 is less than x, which is less than negative 3

  2. negative 3 is less than x, which is less than 1

  3. 1 is less than x, which is less than 3

  4. 3 is less than x, which is less than 4

Show Answer Correct Answer: B

Choice B is correct. The graph of a quadratic function in the xy-plane is a parabola. The axis of symmetry of the parabola passes through the vertex of the parabola. Therefore, the vertex of the parabola and the midpoint of the segment between the two x-intercepts of the graph have the same x-coordinate. Since f of negative 3, equals, f of negative 1, which equals 0, the x-coordinate of the vertex is the fraction with numerator negative 3 plus negative 1, and denominator 2, equals negative 2. Of the shown intervals, only the interval in choice B contains –2. Choices A, C, and D are incorrect and may result from either calculation errors or misidentification of the graph’s x-intercepts.

Question 246 246 of 479 selected Equivalent Expressions H

If 48c=473, what is the value of c ?

Show Answer Correct Answer: .2916, .2917, 7/24

The correct answer is 7 24 . An expression of the form amn, where m and n are integers greater than 1 and a0, is equivalent to amn. Therefore, the expression on the right-hand side of the given equation, 473, is equivalent to 473. Thus, 48c=473. It follows that 8 c = 7 3 . Dividing both sides of this equation by 8 yields c = 7 24 . Note that 7/24, .2916, .2917, 0.219, and 0.292 are examples of ways to enter a correct answer.

Question 247 247 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

Equation 1: x plus y equals 12. Equation 2: y equals x squared.

If the ordered pair x comma y is a solution to the system of equations above, which of the following is a possible value of x?

  1. 0

  2. 1

  3. 2

  4. 3

Show Answer Correct Answer: D

Choice D is correct. Substituting x squared from the second equation for y in the first equation yields x plus x squared, equals 12. Subtracting 12 from both sides of this equation and rewriting the equation results in x squared, plus x, minus 12, equals 0. Factoring the left-hand side of this equation yields open parenthesis, x minus 3, close parenthesis, times, open parenthesis, x plus 4, close parenthesis, equals 0. Using the zero product property to solve for x, it follows that x minus 3, equals 0 and x plus 4, equals 0. Solving each equation for x yields x equals 3 and x equals negative 4, respectively. Thus, two possible values of x are 3 and negative 4. Of the choices given, 3 is the only possible value of x.

Choices A, B, and C are incorrect. Substituting 0 for x in the first equation gives 0 plus y, equals 12, or y equals 12; then, substituting 12 for y and 0 for x in the second equation gives 12 equals 0 squared, or 12 equals 0, which is false. Similarly, substituting 1 or 2 for x in the first equation yields y equals 11 or y equals 10, respectively; then, substituting 11 or 10 for y in the second equation yields a false statement.

 

Question 248 248 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

y = - 2.5

y = x 2 + 8 x + k

In the given system of equations, k is a positive integer constant. The system has no real solutions. What is the least possible value of k ?

Show Answer Correct Answer: 14

The correct answer is 14 . It's given by the first equation of the system of equations that y = - 2.5 . Substituting - 2.5 for y in the second given equation, y=x2+8x+k, yields -2.5=x2+8x+k. Adding 2.5 to both sides of this equation yields 0=x2+8x+k+2.5. A quadratic equation of the form 0=ax2+bx+c, where a , b , and c are constants, has no real solutions if and only if its discriminant, b2-4ac, is negative. In the equation 0=x2+8x+k+2.5, where k is a positive integer constant, a = 1 , b = 8 , and c = k + 2.5 . Substituting 1 for a , 8 for b , and k + 2.5 for c in b2-4ac yields 82-4(1)(k+2.5), or 64-4(k+2.5). Since this value must be negative, 64-4(k+2.5)<0. Adding 4(k+2.5) to both sides of this inequality yields 64<4(k+2.5). Dividing both sides of this inequality by 4 yields 16<k+2.5. Subtracting 2.5 from both sides of this inequality yields 13.5<k. Since k is a positive integer constant, the least possible value of k is 14 .

Question 249 249 of 479 selected Nonlinear Functions H

The function f is defined by f(x)=ax+b, where a and b are constants and a>0. In the xy-plane, the graph of y=f(x) has a y-intercept at (0,-25) and passes through the point (2,23). What is the value of a + b ?

Show Answer Correct Answer: -19

The correct answer is - 19 . It's given that function f is defined by f(x)=ax+b, where a and b are constants and a>0. It's also given that the graph of y=f(x) in the xy-plane has a y-intercept at (0,-25) and passes through the point (2,23). Since the graph has a y-intercept at (0,-25), f(0)=-25. Substituting 0 for x in the given equation yields f(0)=a0+b, or f(0)=1+b, and substituting - 25 for f(0) in this equation yields -25=1+b. Subtracting 1 from each side of this equation yields -26=b. Substituting - 26 for b in the equation f(x)=ax+b yields f(x)=ax-26. Since the graph also passes through the point (2,23), f(2)=23. Substituting 2 for x in the equation f(x)=ax-26 yields f(2)=a2-26, and substituting 23 for f(2) yields 23=a2-26. Adding 26 to each side of this equation yields 49=a2. Taking the square root of both sides of this equation yields ±7=a. Since it's given that a>0, the value of a is 7 . It follows that the value of a + b is 7-26, or - 19 .

Question 250 250 of 479 selected Nonlinear Functions E

12345678910x12345678910yO
  • The parabola opens upward.
  • The vertex is at the point (4 comma 2).
  • The parabola passes through the following points:
    • (0 comma 8)
    • (4 comma 2)
    • (8 comma 8)

The graph shows a marble's height above the ground y , in inches, x seconds after it started moving on an elevated track of a marble run. Which of the following is the best interpretation of the y -intercept of the graph?

  1. The marble's height was 0 inches above the ground 8 seconds after it started moving.

  2. The marble's height was 8 inches above the ground when it started moving.

  3. The marble's minimum height was 0 inches above the ground.

  4. The marble's minimum height was 8 inches above the ground.

Show Answer Correct Answer: B

Choice B is correct. The y-intercept of a graph is the point at which the graph intersects the y-axis. The graph shown intersects the y-axis at the point (0,8). Therefore, the y-intercept of the graph is (0,8). It’s given that y is the height of the marble above the ground, in inches, and x is the number of seconds after the marble started moving. It follows that the marble's height was 8 inches above the ground 0 seconds after it started moving. Therefore, the best interpretation of the y-intercept of the graph is that the marble’s height was 8 inches above the ground when it started moving.

Choice A is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 251 251 of 479 selected Nonlinear Functions M

f of theta equals, negative 0 point 2 8, times, open parenthesis, theta minus 27, close parenthesis, squared, plus 880

An engineer wanted to identify the best angle for a cooling fan in an engine in order to get the greatest airflow. The engineer discovered that the function above models the airflow f of theta, in cubic feet per minute, as a function of the angle of the fan theta, in degrees. According to the model, what angle, in degrees, gives the greatest airflow?

  1. –0.28

  2. 0.28

  3. 27

  4. 880

Show Answer Correct Answer: C

Choice C is correct. The function f is quadratic, so it will have either a maximum or a minimum at the vertex of the graph. Since the coefficient of the quadratic term (–0.28) is negative, the vertex will be at a maximum. The equation f(theta) = –0.28(theta – 27)2 + 880 is given in vertex form, so the vertex is at theta = 27. Therefore, an angle of 27 degrees gives the greatest airflow.

Choices A and B are incorrect and may be the result of misidentifying which value in a quadratic equation in vertex form represents the vertex. Choice D is incorrect. This choice identifies the maximum value of f(theta) rather than the value of theta for which f(theta) is maximized.

 

Question 252 252 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

p = 20 + 16 n

The given equation relates the numbers p and n , where n is not equal to 0 and p>20. Which equation correctly expresses n in terms of p ?

  1. n=p-2016

  2. n = p 16 + 20

  3. n = p 16 - 20

  4. n = 16 p - 20

Show Answer Correct Answer: D

Choice D is correct. To express n in terms of p , the given equation must be solved for n . Subtracting 9 from both sides of the given equation yields p-9=14n. Since n is not equal to 0 , multiplying both sides of this equation by n yields (p-9)(n)=14. It's given that p>9, which means p - 9 is not equal to 0 . Therefore, dividing both sides of (p-9)(n)=14 by (p-9) yields (p-9)(n)p-9=14p-9, or n=14p-9.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 253 253 of 479 selected Nonlinear Functions M

The function f(t)=60,000(2)t410 gives the number of bacteria in a population t minutes after an initial observation. How much time, in minutes, does it take for the number of bacteria in the population to double?

Show Answer Correct Answer: 410

The correct answer is 410 . It's given that t minutes after an initial observation, the number of bacteria in a population is 60,000(2)t410. This expression consists of the initial number of bacteria, 60,000, multiplied by the expression 2t410. The time it takes for the number of bacteria to double is the increase in the value of t that causes the expression 2t410 to double. Since the base of the expression 2t410 is 2 , the expression 2t410 will double when the exponent increases by 1 . Since the exponent of the expression 2t410 is t410, the exponent will increase by 1 when t increases by 410 . Therefore the time, in minutes, it takes for the number of bacteria in the population to double is 410 .

Question 254 254 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

y=(x-2)(x+4)

y=6x-12

Which ordered pair (x,y) is the solution to the given system of equations?

  1. (0,2)

  2. (-4,2)

  3. (2,0)

  4. (2,-4)

Show Answer Correct Answer: C

Choice C is correct. The second equation in the given system of equations is y = 6 x - 12 . Substituting 6 x - 12 for y in the first equation of the given system yields 6x-12=(x-2)(x+4). Factoring 6 out of the left-hand side of this equation yields 6(x-2)=(x-2)(x+4). An expression with a factor of the form (x-a) is equal to zero when x = a . Each side of this equation has a factor of (x-2), so each side of the equation is equal to zero when x = 2 . Substituting 2 for x into the equation 6(x-2)=(x-2)(x+4) yields 6(2-2)=(2-2)(2+4), or 0=0, which is true. Substituting 2 for x into the second equation in the given system of equations yields y=6(2)-12, or y = 0 . Therefore, the solution to the system of equations is the ordered pair (2,0).

Choice A is incorrect and may result from switching the order of the solutions for x and y .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 255 255 of 479 selected Equivalent Expressions E

Which expression is equivalent to 4(x2+6)?

  1. 4 x 2 + 24

  2. 4 x 2 + 10

  3. 4 x 2 + 6

  4. 4 x 2 - 2

Show Answer Correct Answer: A

Choice A is correct. The expression 4(x2+6) can be rewritten as 4(x2)+4(6), which is equivalent to 4x2+24.

Choice B is incorrect. This expression is equivalent to 4(x2+52), not 4(x2+6).

Choice C is incorrect. This expression is equivalent to 4(x2+32), not 4(x2+6).

Choice D is incorrect. This expression is equivalent to 4(x2-12), not 4(x2+6).

Question 256 256 of 479 selected Equivalent Expressions M

Which expression is equivalent to 6 x 8 y 2 + 12 x 2 y 2 ?

  1. 6x2y2(2x6)

  2. 6x2y2(x4)

  3. 6 x 2 y 2 ( x 6 + 2 )

  4. 6 x 2 y 2 ( x 4 + 2 )

Show Answer Correct Answer: C

Choice C is correct. Since each term of the given expression has a common factor of 6x2y2, it may be rewritten as 6x2y2(x6)+6x2y2(2), or 6x2y2(x6+2).

Choice A is incorrect. This expression is equivalent to 12x8y2, not 6x8y2+12x2y2.

Choice B is incorrect. This expression is equivalent to 6x6y2, not 6x8y2+12x2y2.

Choice D is incorrect. This expression is equivalent to 6x6y2+12x2y2, not 6x8y2+12x2y2.

Question 257 257 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

| x - 9 | + 45 = 63

What is the sum of the solutions to the given equation?

Show Answer Correct Answer: 18

The correct answer is 18 . Subtracting 45 from each side of the given equation yields |x-9|=18. By the definition of absolute value, if |x-9|=18, then x - 9 = 18 or x - 9 = -18 . Adding 9 to each side of the equation x - 9 = 18 yields x = 27 . Adding 9 to each side of the equation x - 9 = -18 yields x = -9 . Therefore, the solutions to the given equation are 27 and -9 , and it follows that the sum of the solutions to the given equation is 27+(-9), or 18 .

Question 258 258 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

17b=11xy

The given equation relates the positive numbers b , x , and y . Which equation correctly expresses x in terms of b and y ?

  1. x=7by11

  2. x = y - 77 b

  3. x = y 77 b

  4. x = 77 b y

Show Answer Correct Answer: C

Choice C is correct. Multiplying each side of the given equation by y yields the equivalent equation y7b=11x. Dividing each side of this equation by 11 yields y77b=x, or x=y77b.

Choice A is incorrect. This equation is not equivalent to the given equation. 

Choice B is incorrect. This equation is not equivalent to the given equation. 

Choice D is incorrect. This equation is not equivalent to the given equation. 

Question 259 259 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

(5x+4)(2x-5)=0

Which of the following is a solution to the given equation?

  1. - 5 2

  2. - 5 4

  3. - 4 5

  4. - 2 5

Show Answer Correct Answer: C

Choice C is correct. Since a product of two factors is equal to 0 if and only if at least one of the factors is 0 , either 5 x + 4 = 0 or 2 x - 5 = 0 . Subtracting 4 from each side of the equation 5 x + 4 = 0 yields 5 x = - 4 . Dividing each side of this equation by 5 yields x=-45. Adding 5 to each side of the equation 2 x - 5 = 0 yields 2 x = 5 . Dividing each side of this equation by 2 yields x=52. It follows that the solutions to the given equation are -45 and 52. Therefore, -45 is a solution to the given equation.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 260 260 of 479 selected Nonlinear Functions M
Time (years) Total amount (dollars)
0 670.00
1 674.02
2 678.06

Sara opened a savings account at a bank. The table shows the exponential relationship between the time t , in years, since Sara opened the account and the total amount d , in dollars, in the account. If Sara made no additional deposits or withdrawals, which of the following equations best represents the relationship between t and d ?

  1. d=0.006(1+670)t

  2. d=670(1+0.006)t

  3. d=(1+0.006)t

  4. d=(1+670)t

Show Answer Correct Answer: B

Choice B is correct. It’s given that the relationship between t and d is exponential. The table shows that the value of d increases as the value of t increases. Therefore, the relationship between t and d can be represented by an increasing exponential equation of the form d=a(1+b)t, where a and b are positive constants. The table shows that when t = 0 , d = 670 . Substituting 0 for t and 670 for d in the equation d=a(1+b)t yields 670=a(1+b)0, which is equivalent to 670=a(1), or 670=a. Substituting 670 for a in the equation d=a(1+b)t yields d=670(1+b)t. The table also shows that when t = 1 , d = 674.02 . Substituting 1 for t and 674.02 for d in the equation d=670(1+b)t yields 674.02=670(1+b)1, or 674.02=670(1+b). Dividing both sides of this equation by 670 yields 1.006=1+b. Subtracting 1 from both sides of this equation yields b = 0.006 . Substituting 0.006 for b in the equation d=670(1+b)t yields d=670(1+0.006)t. Therefore, of the choices, choice B best represents the relationship between t and d .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 261 261 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

|p|+61=65

Which value is a solution to the given equation?

  1. 65 61

  2. 4

  3. 126

  4. 130

Show Answer Correct Answer: B

Choice B is correct. Subtracting 61 from each side of the given equation yields |p|=4. By the definition of absolute value, if |p|=4, then p = 4 or p = - 4 . Of the given choices, 4 is a solution to the given equation.

Choice A is incorrect. This is the quotient, not the difference, of 65 and 61 .

Choice C is incorrect. This is the sum, not the difference, of 65 and 61 .

Choice D is incorrect and may result from conceptual or calculation errors.

Question 262 262 of 479 selected Nonlinear Functions H

A landscaper is designing a rectangular garden. The length of the garden is to be 5 feet longer than the width. If the area of the garden will be 104 square feet, what will be the length, in feet, of the garden?

Show Answer

The correct answer is 13. Let w represent the width of the rectangular garden, in feet. Since the length of the garden will be 5 feet longer than the width of the garden, the length of the garden will be w plus 5 feet. Thus the area of the garden will be w times, open parenthesis, w plus 5, close parenthesis. It is also given that the area of the garden will be 104 square feet. Therefore, w times, open parenthesis, w plus 5, close parenthesis, equals 104, which is equivalent to w squared, plus 5 w, minus 104, equals 0. Factoring this equation results in open parenthesis, w plus 13, close parenthesis, times, open parenthesis, w minus 8, close parenthesis, equals 0. Therefore, w equals 8 and w equals negative 13. Because width cannot be negative, the width of the garden must be 8 feet. This means the length of the garden must be 8 plus 5, equals 13 feet.

Question 263 263 of 479 selected Nonlinear Functions H

P(t)=290(1.04)(46)t

The function P models the population, in thousands, of a certain city t years after 2005 . According to the model, the population is predicted to increase by n% every 18 months. What is the value of n ?

  1. 0.38

  2. 1.04

  3. 4

  4. 6

Show Answer Correct Answer: C

Choice C is correct. It's given that the function P models the population of the city t years after 2005. Since there are 12 months in a year, 18 months is equivalent to 1812 years. Therefore, the expression 1812x can represent the number of years in x 18 -month periods. Substituting 1812x for t in the given equation yields P(1812x)=290(1.04)(46)(1812x), which is equivalent to P(1812x)=290(1.04)x. Therefore, for each 18 -month period, the predicted population of the city is 1.04 times, or 104% of, the previous population. This means that the population is predicted to increase by 4% every 18 months.

Choice A is incorrect and may result from conceptual or calculation errors. 

Choice B is incorrect. Each year, the predicted population of the city is 1.04 times the previous year's predicted population, which is not the same as an increase of 1.04%.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 264 264 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

  • For the parabola in the system:
    • The parabola opens upward.
    • The vertex is at point (4 comma 1).
    • The parabola passes through the following points:
      • (3 comma 2)
      • (4 comma 1)
      • (5 comma 2)
  • For the line in the system:
    • The line is horizontal.
    • The line passes through the following points:
      • (0 comma 1)
      • (4 comma 1)

The graph of a system of a linear and a quadratic equation is shown. What is the solution (x,y) to this system?

  1. (0,0)

  2. (-4,1)

  3. (4,-1)

  4. (4,1)

Show Answer Correct Answer: D

Choice D is correct. The solution to the system corresponds to the point where the graphs of the equations intersect. The graphs of the linear equation and the quadratic equation shown intersect at the point (4,1). Therefore, (4,1) is the solution to this system.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 265 265 of 479 selected Equivalent Expressions E

Which of the following is equivalent to 2 times, open parenthesis, x squared minus x, close parenthesis, plus 3 times, open parenthesis, x squared minus x, close parenthesis ?

  1. 5 x squared, minus 5 x

  2. 5 x squared, plus 5 x

  3. 5x

  4. 5x2

Show Answer Correct Answer: A

Choice A is correct. Since open parenthesis, x squared, minus x, close parenthesis is a common term in the original expression, like terms can be added: 2 times, open parenthesis, x squared, minus x, close parenthesis, plus, 3 times, open parenthesis, x squared, minus x, close parenthesis, equals, 5 times, open parenthesis, x squared, minus x, close parenthesis. Distributing the constant term 5 yields 5 x squared, minus 5 x.

Choice B is incorrect and may result from not distributing the negative signs in the expressions within the parentheses. Choice C is incorrect and may result from not distributing the negative signs in the expressions within the parentheses and from incorrectly eliminating the x squared-term. Choice D is incorrect and may result from incorrectly eliminating the x-term.

 

Question 266 266 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

7m=2(n+p)

The given equation relates the positive numbers m , n , and p . Which equation correctly gives m in terms of n and p ?

  1. m=2(n+p)7

  2. m=2(n+p)

  3. m=2(n+p)-7

  4. m=2-n-p-7

Show Answer Correct Answer: A

Choice A is correct. Dividing each side of the given equation by 7 yields m=2(n+p)7.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This equation is equivalent to 7+m=2(n+p), not 7m=2(n+p).

Choice D is incorrect and may result from conceptual or calculation errors.

Question 267 267 of 479 selected Nonlinear Functions M

The function f is defined by f(x)=8x3+4. What is the value of f(2)?

Show Answer Correct Answer: 68

The correct answer is 68 . It's given that the function f is defined by f(x)=8x3+4. Substituting 2 for x in this equation yields f(2)=8(2)3+4, or f(2)=8(8)+4, which is equivalent to f(2)=68. Therefore, the value of f(2) is 68 .

Question 268 268 of 479 selected Equivalent Expressions E

Which expression is equivalent to 13x2-7x2?

  1. -91 x 2

  2. 6 x 2

  3. 20 x 2

  4. 40 x 2

Show Answer Correct Answer: B

Choice B is correct. Since each term in the given expression has a common factor of x2, it can be rewritten as (13-7)x2, or 6x2. Therefore, the expression 6x2 is equivalent to 13x2-7x2.

Alternate approach: Since the two terms of the given expression are both constant multiples of x2, they are like terms and can, therefore, be combined through subtraction. Subtracting like terms in the expression 13x2-7x2 yields 6x2.

Choice A is incorrect. This expression is equivalent to (13x)(-7x), not 13x2-7x2.

Choice C is incorrect. This expression is equivalent to 13x2+7x2, not 13x2-7x2.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 269 269 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

In the xy-plane, the graph of y equals, x squared minus 9 intersects line p at the point with coordinates 1 comma a and the point with coordinates 5 comma b, where a and b are constants. What is the slope of line p ?

  1. 6

  2. 2

  3. negative 2

  4. negative 6

Show Answer Correct Answer: A

Choice A is correct. It’s given that the graph of y equals, x squared, minus 9 and line p intersect at the point with coordinates 1 comma a and the point with coordinates 5 comma b. Therefore, the value of y when x equals 1 is the value of a, and the value of y when x equals 5  is the value of b. Substituting 1 for x in the given equation yields y equals, 1 squared minus 9, or y equals negative 8. Similarly, substituting 5 for x in the given equation yields y equals, 5 squared minus 9, or y equals 16. Therefore, the intersection points are the point with coordinates 1 comma negative 8 and the point with coordinates 5 comma 16. The slope of line p is the ratio of the change in y to the change in x between these two points: the fraction with numerator 16 minus negative 8, and denominator 5 minus 1, end fraction, equals the fraction 24 over 4, or 6.

Choices B, C, and D are incorrect and may result from conceptual or calculation errors in determining the values of a, b, or the slope of line p.

 

Question 270 270 of 479 selected Equivalent Expressions M

Which expression is equivalent to a 11 12 , where a>0?

  1. a 132 12

  2. a 132 144

  3. a 132 121

  4. a 132 11

Show Answer Correct Answer: B

Choice B is correct. Since 1212=1, multiplying the exponent of the given expression by 1212 yields an equivalent expression: a(1112)(1212)=a(132144). Since 132144=132(1144), the expression a132144 can be rewritten as a(132)(1144). Applying properties of exponents, this expression can be rewritten as (a132)1144. An expression of the form (m)1k, where m>0 and k>0, is equivalent to mk. Therefore, (a132)1144 is equivalent to a132144.

Choice A is incorrect and may result from conceptual or calculation errors. 

Choice C is incorrect and may result from conceptual or calculation errors. 

Choice D is incorrect and may result from conceptual or calculation errors. 

Question 271 271 of 479 selected Equivalent Expressions E

Which of the following is equivalent to 2 x cubed, plus 4 ?

  1. 4 times open parenthesis, x cubed, plus 4, close parenthesis

  2. 4 times open parenthesis, x cubed, plus 2, close parenthesis

  3. 2 times open parenthesis, x cubed, plus 4, close parenthesis

  4. 2 times open parenthesis, x cubed, plus 2, close parenthesis

Show Answer Correct Answer: D

Choice D is correct. The expression 2 x cubed, plus 4 has two terms, 2 x cubed and 4. The greatest common factor of these two terms is 2. Factoring 2 from each of these terms yields 2 times x cubed, plus, 2 times 2, or 2 times, open parenthesis, x cubed plus 2, close parenthesis.

Choices A and B are incorrect because 4 is not a factor of the term 2 x cubed. Choice C is incorrect and may result from factoring 2 from 2 x cubed but not from 4.

 

Question 272 272 of 479 selected Nonlinear Functions H

For the function f , f(0)=86, and for each increase in x by 1 , the value of f(x) decreases by 80%. What is the value of f(2)?

Show Answer Correct Answer: 3.44, 86/25

The correct answer is 3.44 . It’s given that f(0)=86 and that for each increase in x by 1 , the value of f(x) decreases by 80%. Because the output of the function decreases by a constant percentage for each 1 -unit increase in the value of x , this relationship can be represented by an exponential function of the form f(x)=a(b)x, where a represents the initial value of the function and b represents the rate of decay,
expressed as a decimal. Because f(0)=86, the value of a must be 86 . Because the value of f(x) decreases by 80% for each 1 -unit increase in x , the value of b must be (10.80), or 0.2. Therefore, the function f  can be defined by f(x)=86(0.2)x. Substituting 2 for x in this function yields f(2)=86(0.2)2, which is equivalent to f(2)=86(0.04), or f(2)=3.44. Either 3.44 or 86/25 may be entered as the correct answer.

Alternate approach: It’s given that f(0)=86 and that for each increase in x by 1 , the value of f(x) decreases by 80%. Therefore, when x=1, the value of f(x) is (10080)%, or 20%, of 86 , which can be expressed as (0.20)(86). Since (0.20)(86)=17.2, the value of f(1) is 17.2 . Similarly, when x=2, the value of f(x) is 20% of 17.2 , which can be expressed as (0.20)(17.2). Since (0.20)(17.2)=3.44, the value of f(2) is 3.44 . Either 3.44 or 86/25 may be entered as the correct answer.

Question 273 273 of 479 selected Nonlinear Functions M

  • The parabola opens downward.
  • The vertex is at the approximate point (0.2 comma 10.2).
  • The parabola passes through the following points:
    • (0 comma 10)
    • approximately (0.2 comma 10.2)
    • approximately (1 comma 6.9)
    • approximately (1.6 comma 0)

A competitive diver dives from a platform into the water. The graph shown gives the height above the water y , in meters, of the diver x seconds after diving from the platform. What is the best interpretation of the x-intercept of the graph?

  1. The diver reaches a maximum height above the water at 1.6 seconds.

  2. The diver hits the water at 1.6 seconds.

  3. The diver reaches a maximum height above the water at 0.2 seconds.

  4. The diver hits the water at 0.2 seconds.

Show Answer Correct Answer: B

Choice B is correct. It’s given that the graph shows the height above the water y , in meters, of a diver x seconds after diving from a platform. The x-intercept of a graph is the point at which the graph intersects the x-axis, or when the value of y is 0 . The graph shown intersects the x-axis between x = 1 and x = 2 . In other words, the diver is 0 meters above the water, or hits the water, between 1 and 2 seconds after diving from the platform. Of the given choices, only choice B includes an interpretation where the diver hits the water between 1 and 2 seconds. Therefore, the best interpretation of the x-intercept of the graph is the diver hits the water at 1.6 seconds.

Choice A is incorrect and may result from conceptual errors.

Choice C is incorrect. This is the best interpretation of the maximum value, not the x-intercept, of the graph.

Choice D is incorrect and may result from conceptual errors.

Question 274 274 of 479 selected Nonlinear Functions E

The function h is defined by h(x)=85x+6. What is the value of h(2)?

Show Answer Correct Answer: .5, 1/2

The correct answer is 12. The value of h(2) is the value of h(x) when x = 2 . Substituting 2 for x in the given equation yields h(2)=85(2)+6, which is equivalent to h(2)=816, or h(2)=12. Therefore, the value of h(2) is 12. Note that 1/2 and .5 are examples of ways to enter a correct answer.

Question 275 275 of 479 selected Nonlinear Functions H

M equals 1,800 times, 1 point 0 2, to the power t

The equation above models the number of members, M, of a gym t years after the gym opens. Of the following, which equation models the number of members of the gym q quarter years after the gym opens?

  1. M equals 1,800 times, 1 point 0 2, to the power of the fraction q over 4

  2. M equals 1,800 times, 1 point 0 2, to the power 4 q

  3. M equals 1,800 times, 1 point 0 0 5, to the power 4 q

  4. M equals 1,800 times, 1 point 0 8 2, to the power q

Show Answer Correct Answer: A

Choice A is correct. In 1 year, there are 4 quarter years, so the number of quarter years, q, is 4 times the number of years, t ; that is, q, equals 4, t. This is equivalent to  t equals, q over 4, and substituting this into the expression for M in terms of t gives M equals, 1,800 times, 1 point 0 2 raised to the fraction q over 4 power.

Choices B and D are incorrect and may be the result of incorrectly using t equals, 4 q instead of q, equals 4, t. (Choices B and D are nearly the same since 1 point 0 2 raised to the 4 q power is equivalent to open parenthesis, 1 point 0 2 to the fourth power, close parenthesis, raised to the q power, which is approximately 1 point 0 8 2 to the q power.) Choice C is incorrect and may be the result of incorrectly using t equals, 4 q and unnecessarily dividing 0.02 by 4.

 

Question 276 276 of 479 selected Nonlinear Functions M

A certain college had 3,000 students enrolled in 2015. The college predicts that after 2015, the number of students enrolled each year will be 2% less than the number of students enrolled the year before. Which of the following functions models the relationship between the number of students enrolled, f of x, and the number of years after 2015, x ?

  1. f of x equals, 0 point 0 2, times open parenthesis, 3,000, close parenthesis, to the x power

  2. f of x equals, 0 point 9 8, times open parenthesis, 3,000, close parenthesis, to the x power

  3. f of x equals, 3,000, times open parenthesis, 0 point 0 2, close parenthesis, to the x power

  4. f of x equals, 3,000, times open parenthesis, 0 point 9 8, close parenthesis, to the x power,

Show Answer Correct Answer: D

Choice D is correct. Because the change in the number of students decreases by the same percentage each year, the relationship between the number of students and the number of years can be modeled with a decreasing exponential function in the form f of x equals, a, times, open parenthesis, 1 minus r, close parenthesis, to the x power , where f of x is the number of students, a is the number of students in 2015, r is the rate of decrease each year, and x is the number of years since 2015. It’s given that 3,000 students were enrolled in 2015 and that the rate of decrease is predicted to be 2%, or 0.02. Substituting these values into the decreasing exponential function yields f of x equals, 3,000 times, open parenthesis, 1 minus 0 point 0 2, close parenthesis, to the x power, which is equivalent to f of x equals, 3,000 times, open parenthesis, 0 point 9 8, close parenthesis, to the x power.

Choices A, B, and C are incorrect and may result from conceptual errors when translating the given information into a decreasing exponential function.

 

Question 277 277 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

y = x + 9

y = x 2 + 16 x + 63

A solution to the given system of equations is (x,y). What is the greatest possible value of x ?

  1. -6

  2. 7

  3. 9

  4. 63

Show Answer Correct Answer: A

Choice A is correct. It's given that y = x + 9 and y = x 2 + 16 x + 63 ; therefore, it follows that x + 9 = x 2 + 16 x + 63 . This equation can be rewritten as x+9=(x+9)(x+7). Subtracting (x+9) from both sides of this equation yields 0=(x+9)(x+7)-(x+9). This equation can be rewritten as 0=(x+9)((x+7)-1), or 0=(x+9)(x+6). By the zero product property, x + 9 = 0 or x + 6 = 0 . Subtracting 9 from both sides of the equation x + 9 = 0 yields x = -9 . Subtracting 6 from both sides of the equation x + 6 = 0 yields x = -6 . Therefore, the given system of equations has solutions, (x,y), that occur when x = -9 and x = -6 . Since -6 is greater than -9 , the greatest possible value of x is -6 .

Choice B is incorrect. This is the negative of the greatest possible value of x when y = 0 for the second equation in the given system of equations.

Choice C is incorrect. This is the value of y when x = 0 for the first equation in the given system of equations.

Choice D is incorrect. This is the value of y when x = 0 for the second equation in the given system of equations.

Question 278 278 of 479 selected Nonlinear Functions E

The function f(x)=200,000(1.21)x gives a company’s predicted annual revenue, in dollars, x years after the company started selling light bulbs online, where 0<x10. What is the best interpretation of the statement “f(5) is approximately equal to 518,748 ” in this context?

  1. 5 years after the company started selling light bulbs online, its predicted annual revenue is approximately 518,748 dollars.

  2. 5 years after the company started selling light bulbs online, its predicted annual revenue will have increased by a total of approximately 518,748 dollars.

  3. When the company’s predicted annual revenue is approximately 518,748 dollars, it is 5 times the predicted annual revenue for the previous year.

  4. When the company’s predicted annual revenue is approximately 518,748 dollars, it is 5 % greater than the predicted annual revenue for the previous year.

Show Answer Correct Answer: A

Choice A is correct. It's given that the function f(x)=200,000(1.21)x gives a company's predicted annual revenue, in dollars, x years after the company started selling light bulbs online. It follows that f(x) represents the company's predicted annual revenue, in dollars, x years after the company started selling light bulbs online. Since the value of f(5) is the value of f(x) when x=5, it follows that "f(5) is approximately equal to 518,748" means that f(x) is approximately equal to 518,748 when x=5. Therefore, the best interpretation of the statement "f(5) is approximately equal to 518,748" in this context is 5 years after the company started selling light bulbs online, its predicted annual revenue is approximately 518,748 dollars.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 279 279 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

(x+9)(x-9)x+9=7

What is the solution to the given equation?

  1. 7

  2. 9

  3. 16

  4. 63

Show Answer Correct Answer: C

Choice C is correct. Since the left-hand side of the given equation has a factor of x + 9 in both the numerator and the denominator, the solution to the given equation can be found by solving the equation x - 9 = 7 . Adding 9 to both sides of this equation yields x = 16 . Substituting 16 for x in the given equation yields (16+9)(16-9)16+9=7, or 7=7. Therefore, the solution to the given equation is 16 .

Choice A is incorrect. Substituting 7 for x in the given equation yields (7+9)(7-9)7+9=7, or -2=7, which is false.

Choice B is incorrect. Substituting 9 for x in the given equation yields (9+9)(9-9)9+9=7, or 0=7, which is false.

Choice D is incorrect. Substituting 63 for x in the given equation yields (63+9)(63-9)63+9=7, or 54=7, which is false.

Question 280 280 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

20p=20q-20r-20s

The given equation relates the positive variables p , q , r , and s . Which of the following is equivalent to q ?

  1. p+r+s

  2. 20(p+r+s)

  3. prspr+ps+rs

  4. prs20p+20r+20s

Show Answer Correct Answer: C

Choice C is correct. Multiplying each side of the given equation by 120 yields 120(20p)=120(20q-20r-20s). Distributing 120 on each side of this equation yields 2020p=2020q-2020r-2020s, or 1p=1q-1r-1s. Adding 1r+1s to each side of this equation yields 1s+1r+1p=1q. Multiplying 1s by prpr, 1r by psps, and 1p by rsrs yields prprs+psprs+rsprs=1q, which is equivalent to pr+ps+rsprs=1q. Since pr+ps+rsprs=1q, and it's given that p , q , r , and s are positive, it follows that the reciprocals of each side of this equation are also equal. Thus, prspr+ps+rs=q1, or prspr+ps+rs=q. Therefore, prspr+ps+rs is equivalent to q .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 281 281 of 479 selected Equivalent Expressions M

The sum of negative 2 x squared, plus x, plus 31 and 3 x squared, plus 7 x, minus 8 can be written in the form a, x squared, plus b x, plus c, where a, b, and c are constants. What is the value of a, plus b plus c ?

Show Answer

The correct answer is 32. The sum of the given expressions is open parenthesis, negative 2, x squared, plus x, plus 31, close parenthesis, plus, open parenthesis, 3 x squared, plus 7 x, minus 8, close parenthesis. Combining like terms yields x squared, plus 8 x, plus 23. Based on the form of the given equation, a, equals 1, b equals 8, and c equals 23. Therefore, a, plus b, plus c, equals 32.

Alternate approach: Because a, plus b, plus c is the value of a, x squared, plus b x, plus c when x equals 1, it is possible to first make that substitution into each polynomial before adding them. When x equals 1, the first polynomial is equal tonegative 2 plus 1, plus 31, equals 30 and the second polynomial is equal to 3 plus 7, minus 8, equals 2. The sum of 30 and 2 is 32.

Question 282 282 of 479 selected Equivalent Expressions M

open parenthesis, one half x, plus three halves, close parenthesis, times, open parenthesis, three halves x, plus one half, close parenthesis

The expression above is equivalent to a, x squared, plus b x, plus c , where a, b, and c are constants. What is the value of b?

Show Answer

The correct answer is five halves. The expression open parenthesis, one half x, plus three halves, close parenthesis, times, open parenthesis, three halves x, plus one half, close parenthesis can be written in the form a, x squared, plus b x, plus c, where a, b, and c are constants, by multiplying out the expression using the distributive property of multiplication over addition. The result is one half x, times three halves x, plus, one half x, times one half, plus, three halves times three halves x, plus, three halves times one half. This expression can be rewritten by multiplying as indicated to give three fourths x squared, plus, one fourth x, plus, nine fourths x, plus, three fourths, which can be simplified to three fourths x squared, plus, ten fourths x, plus, three fourths, or three fourths x squared, plus, five halves x, plus, three fourths. This is in the form a, x squared, plus b x, plus c, where the value of b is five halves. Note that 5/2 and 2.5 are examples of ways to enter a correct answer.

Question 283 283 of 479 selected Nonlinear Functions H

The functions f and g are defined by the given equations, where x0. Which of the following equations displays, as a constant or coefficient, the maximum value of the function it defines, where x0?

  1. f(x)=33(0.4)x+3
  2. g(x)=33(0.16)(0.4)x-2
  1. I only

  2. II only

  3. I and II

  4. Neither I nor II

Show Answer Correct Answer: B

Choice B is correct. Functions f and g are both exponential functions with a base of 0.40 . Since 0.40 is less than 1 , functions f and g are both decreasing exponential functions. This means that f(x) and g(x) decrease as x increases. Since f(x) and g(x) decrease as x increases, the maximum value of each function occurs at the least value of x for which the function is defined. It's given that functions f and g are defined for x0. Therefore, the maximum value of each function occurs at x = 0 . Substituting 0 for x in the equation defining f yields f(0)=33(0.4)0+3, which is equivalent to f(0)=33(0.4)3, or f(0)=2.112. Therefore, the maximum value of f is 2.112 . Since the equation f(x)=33(0.4)x+3 doesn't display the value 2.112 , the equation defining f doesn't display the maximum value of f . Substituting 0 for x in the equation defining g yields g(0)=33(0.16)(0.4)0-2, which can be rewritten as g(0)=33(0.16)(10.42), or g(0)=33(0.16)(10.16), which is equivalent to g(0)=33. Therefore, the maximum value of g is 33 . Since the equation g(x)=33(0.16)(0.4)x-2 displays the value 33 , the equation defining g displays the maximum value of g . Thus, only equation II displays, as a constant or coefficient, the maximum value of the function it defines.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 284 284 of 479 selected Nonlinear Functions H

The function f is defined by f(x)=ax2+bx+c, where a , b , and c are constants. The graph of y=f(x) in the xy-plane passes through the points (7,0) and (-3,0). If a is an integer greater than 1 , which of the following could be the value of a + b ?

  1. -6

  2. -3

  3. 4

  4. 5

Show Answer Correct Answer: A

Choice A is correct. It's given that the graph of y=f(x) in the xy-plane passes through the points (7,0) and (-3,0). It follows that when the value of x is either 7 or -3 , the value of f(x) is 0 . It's also given that the function f is defined by f(x)=ax2+bx+c, where a , b , and c are constants. It follows that the function f is a quadratic function and, therefore, may be written in factored form as f(x)=a(x-u)(x-v), where the value of f(x) is 0 when x is either u or v . Since the value of f(x) is 0 when the value of x is either 7 or -3 , and the value of f(x) is 0 when the value of x is either u or v , it follows that u and v are equal to 7 and -3 . Substituting 7 for u and -3 for v in the equation f(x)=a(x-u)(x-v) yields f(x)=a(x-7)(x-(-3)), or f(x)=a(x-7)(x+3). Distributing the right-hand side of this equation yields f(x)=a(x2-7x+3x-21), or f(x)=ax2-4ax-21a. Since it's given that f(x)=ax2+bx+c, it follows that b=-4a. Adding a to each side of this equation yields a+b=-3a. Since a+b=-3a, if a is an integer, the value of a + b must be a multiple of 3 . If a is an integer greater than 1 , it follows that a2. Therefore, -3a-3(2). It follows that the value of a + b is less than or equal to -3(2), or -6 . Of the given choices, only -6 is a multiple of 3 that's less than or equal to -6 .

Choice B is incorrect. This is the value of a+b if a is equal to, not greater than, 1 .

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 285 285 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

Blood volume,V sub B, in a human can be determined using the equation V sub B equals, the fraction with numerator V sub P, and denominator 1 minus H, end fraction, where V sub P is the plasma volume and H is the hematocrit (the fraction of blood volume that is red blood cells). Which of the following correctly expresses the hematocrit in terms of the blood volume and the plasma volume?

  1. H equals, 1 minus the fraction V sub P over V sub B

  2. H equals, the fraction V sub B over V sub P

  3. H equals, 1 plus the fraction V sub B over V sub P

  4. H equals, V sub B, minus V sub P

Show Answer Correct Answer: A

Choice A is correct. The hematocrit can be expressed in terms of the blood volume and the plasma volume by solving the given equation V sub B equals, the fraction with numerator V sub P, and denominator 1 minus H, end fraction for H. Multiplying both sides of this equation by open parenthesis, 1 minus H, close parenthesis yields V sub B, times, open parenthesis, 1 minus H, close parenthesis, equals V sub P. Dividing both sides by V sub B yields 1 minus H equals, the fraction V sub P, over V sub B. Subtracting 1 from both sides yields negative H equals, negative 1 plus, the fraction V sub P over V sub B. Dividing both sides by negative 1 yields H equals, 1 minus, the fraction V sub P over V sub B.

Choices B, C, and D are incorrect and may result from errors made when manipulating the equation.

 

Question 286 286 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

k 2 - 53 = 91

What is the positive solution to the given equation?

  1. 144

  2. 72

  3. 38

  4. 12

Show Answer Correct Answer: D

Choice D is correct. Adding 53 to each side of the given equation yields k 2 = 144 . Taking the square root of each side of this equation yields k=±12. Therefore, the positive solution to the given equation is 12 .

Choice A is incorrect. This is the positive solution to the equation k 2 - 53 = 20,683 , not k 2 - 53 = 91 .

Choice B is incorrect. This is the positive solution to the equation k 2 - 53 = 5,131 , not k 2 - 53 = 91 .

Choice C is incorrect. This is the positive solution to the equation k 2 - 53 = 1,391 , not k 2 - 53 = 91 .

Question 287 287 of 479 selected Nonlinear Functions M

A model predicts that the population of Bergen was 15,000 in 2005. The model also predicts that each year for the next 5 years, the population p  increased by 4% of the previous year's population. Which equation best represents this model, where x  is the number of years after 2005, for x5?

  1. p=0.96(15,000)x

  2. p=1.04(15,000)x

  3. p=15,000(0.96)x

  4. p=15,000(1.04)x

Show Answer Correct Answer: D

Choice D is correct. It's given that a model predicts the population of Bergen in 2005 was 15,000 . The model also predicts that each year for the next 5 years, the population increased by 4% of the previous year's population. The predicted population in one of these years can be found by multiplying the predicted population from the previous year by 1.04 . Since the predicted population in 2005 was 15,000 , the predicted population 1 year later is 15,000(1.04). The predicted population 2 years later is this value times 1.04 , which is 15,000(1.04)(1.04), or 15,000(1.04)2. The predicted population 3 years later is this value times 1.04 , or 15,000(1.04)3. More generally, the predicted population, p , x years after 2005 is represented by the equation p=15,000(1.04)x

Choice A is incorrect. Substituting 0 for x in this equation indicates the predicted population in 2005 was 0.96 rather than 15,000 .

Choice B is incorrect. Substituting 0 for x in this equation indicates the predicted population in 2005 was 1.04 rather than 15,000 .

Choice C is incorrect. This equation indicates the predicted population is decreasing, rather than increasing, by 4% each year.

Question 288 288 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

-54w=6

What is the solution to the given equation?

Show Answer Correct Answer: -9

The correct answer is -9 . Since w is in the denominator of a fraction in the given equation, w can't be equal to 0 . Since w isn't equal to 0 , multiplying both sides of the given equation by w yields an equivalent equation, -54 = 6 w . Dividing both sides of this equation by 6 yields -9 = w . Therefore, -9 is the solution to the given equation.

Question 289 289 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

P equals, the fraction W over t

The power P produced by a machine is represented by the equation above, where W is the work performed during an amount of time t. Which of the following correctly expresses W in terms of P and t ?

  1. W equals, P t

  2. W equals, the fraction P over t

  3. W equals, the fraction t over P

  4. W equals, P plus t

Show Answer Correct Answer: A

Choice A is correct. Multiplying both sides of the equation by t yields P times t equals, open parenthesis, the fraction w over t, close parenthesis, times t, or P t equals W, which expresses W in terms of P and t. This is equivalent to W = Pt.


Choices B, C, and D are incorrect. Each of the expressions given in these answer choices gives W in terms of P and t but doesn’t maintain the given relationship between W, P, and t. These expressions may result from performing different operations with t on each side of the equation. In choice B, W has been multiplied by t, and P has been divided by t. In choice C, W has been multiplied by t, and the quotient of P divided by t has been reciprocated. In choice D, W has been multiplied by t, and P has been added to t. However, in order to maintain the relationship between the variables in the given equation, the same operation must be performed with t on each side of the equation. 

 

Question 290 290 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

Equation 1: x plus y equals 17. Equation 2: x y equals 72.

If one solution to the system of equations above is the ordered pair x comma y , what is one possible value of x ?

Show Answer

The correct answer is either 8 or 9. The first equation can be rewritten as y equals, 17 minus x. Substituting 17 minus x for y in the second equation gives x times, open parenthesis, 17 minus x, close parenthesis, equals 72. By applying the distributive property, this can be rewritten as 17 x, minus x squared, equals 72. Subtracting 72 from both sides of the equation yields x squared, minus 17 x, plus 72, equals 0. Factoring the left-hand side of this equation yields open parenthesis, x minus 8, close parenthesis, times, open parenthesis, x minus 9, close parenthesis, equals 0. Applying the Zero Product Property, it follows that x minus 8, equals 0 and x minus 9, equals 0. Solving each equation for x yields x equals 8 and x equals 9 respectively. Note that 8 and 9 are examples of ways to enter a correct answer.

Question 291 291 of 479 selected Equivalent Expressions M

Which expression is equivalent to h15q7h5q21, where h>0 and q>0?

  1. h 10 q 14

  2. h 3 q 3

  3. h 10 q 14

  4. h 3 q 3

Show Answer Correct Answer: A

Choice A is correct. For positive values of a , aman=a(m-n), where m and n are integers. Since it's given that h>0 and q>0, this property can be applied to rewrite the given expression as (h(15-5))(q(7-21)), which is equivalent to h10q-14. For positive values of a , a-n=1an. This property can be applied to rewrite the expression h10q-14 as (h10)(1q14), which is equivalent to h10q14.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 292 292 of 479 selected Equivalent Expressions E

Which expression is equivalent to 5 x 5 - 6 x 4 + 8 x 3 ?

  1. x 4 ( 5 x - 6 )

  2. x 3 ( 5 x 2 - 6 x + 8 )

  3. 8 x 3 ( 5 x 2 - 6 x + 1 )

  4. 6 x 5 ( - 6 x 4 + 8 x 3 + 1 )

Show Answer Correct Answer: B

Choice B is correct. Since x 3 is a common factor of each term in the given expression, the expression can be rewritten as x3(5x2-6x+8).

Choice A is incorrect. This expression is equivalent to 5 x 5 - 6 x 4 .

Choice C is incorrect. This expression is equivalent to 40 x 5 - 48 x 4 + 8 x 3 .

Choice D is incorrect. This expression is equivalent to - 36 x 9 + 48 x 8 + 6 x 5 .

Question 293 293 of 479 selected Nonlinear Functions M

A rubber ball bounces upward one-half the height that it falls each time it hits the ground. If the ball was originally dropped from a distance of 20.0 feet above the ground, what was its maximum height above the ground, in feet, between the third and fourth time it hit the ground?

Show Answer

The correct answer is 2.5. After hitting the ground once, the ball bounces to 20 point zero, divided by 2, equals 10 point zero feet. After hitting the ground a second time, the ball bounces to 10 point zero, divided by 2, equals 5 point zero feet. After hitting the ground for the third time, the ball bounces to 5 point zero, divided by 2, equals 2 point 5 feet. Note that 2.5 and 5/2 are examples of ways to enter a correct answer.

Question 294 294 of 479 selected Nonlinear Functions E

The function f is defined by f(x)=6+x. What is the value of f(36)?

Show Answer Correct Answer: 12

The correct answer is 12 . The value of f(36) is the value of f(x) when x=36. Substituting 36 for x in the given equation yields f(36)=6+36, which is equivalent to f(36)=6+6, or f(36)=12. Thus, the value of f(36) is 12 .

Question 295 295 of 479 selected Nonlinear Functions E

The function f(x)=240,000(1.22)x gives a company’s predicted annual revenue, in dollars, x years after the company started selling jewelry online, where 0<x10. What is the best interpretation of the statement “f(5) is approximately equal to 648,650 ” in this context?

  1. 5 years after the company started selling jewelry online, its predicted annual revenue is approximately 648,650 dollars.

  2. 5 years after the company started selling jewelry online, its predicted annual revenue will have increased by a total of approximately 648,650 dollars.

  3. When the company’s predicted annual revenue is approximately 648,650 dollars, it is 5 times the predicted annual revenue for the previous year.

  4. When the company’s predicted annual revenue is approximately 648,650 dollars, it is 5 % greater than the predicted annual revenue for the previous year.

Show Answer Correct Answer: A

Choice A is correct. It’s given that the function f gives a company’s predicted annual revenue, in dollars, x years after the company started selling jewelry online. Since the value of f(5) is the value of f(x) when x=5, it follows that “f(5) is approximately equal to 648,650” means that f(x) is approximately equal to 648,650 when x=5. Therefore, the best interpretation of the given statement is that 5 years after the company started selling jewelry online, its predicted annual revenue is approximately 648,650 dollars.

Choice B is incorrect. The function f gives the predicted annual revenue, not the predicted increase in annual revenue.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect. In the given function, x represents the number of years after the company started selling jewelry online, not the percent increase in revenue from the previous year.

Question 296 296 of 479 selected Equivalent Expressions E

Which expression is equivalent to 9 x 2 + 5 x ?

  1. x ( 9 x + 5 )

  2. 5 x ( 9 x + 1 )

  3. 9 x ( x + 5 )

  4. x 2 ( 9 x + 5 )

Show Answer Correct Answer: A

Choice A is correct. Since x is a factor of each term in the given expression, the expression is equivalent to x(9x)+x(5), or x(9x+5).

Choice B is incorrect. This expression is equivalent to 45 x 2 + 5 x , not 9 x 2 + 5 x .

Choice C is incorrect. This expression is equivalent to 9 x 2 + 45 x , not 9 x 2 + 5 x .

Choice D is incorrect. This expression is equivalent to 9 x 3 + 5 x 2 , not 9 x 2 + 5 x .

Question 297 297 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

 

D equals, T minus, the fraction 9 over 25, end fraction, times open parenthesis, 100 minus H, close parenthesis

The formula above can be used to approximate the dew point D, in degrees Fahrenheit, given the temperature T, in degrees Fahrenheit, and the relative humidity of H percent, where H is greater than 50. Which of the following expresses the relative humidity in terms of the temperature and the dew point?

 

  1. H equals, the fraction 25 over 9, end fraction, times open parenthesis, D minus T, close parenthesis, plus 100

  2. H equals, the fraction 25 over 9, end fraction, times open parenthesis, D minus T, close parenthesis, minus 100

  3. H equals, the fraction 25 over 9, end fraction, times open parenthesis, D plus T, close parenthesis, plus 100

  4. H equals, the fraction 25 over 9, end fraction, times open parenthesis, D plus T, close parenthesis, minus 100

Show Answer Correct Answer: A

Choice A is correct. It’s given that D equals, T minus the fraction 9 over 25, end fraction, times, open parenthesis, 100 minus H, close parenthesis. Solving this formula for H expresses the relative humidity in terms of the temperature and the dew point. Subtracting T from both sides of this equation yields D minus T equals, the negative of the fraction 9 over 25, end fraction, times, open parenthesis, 100 minus H, close parenthesis. Multiplying both sides by the negative of the fraction 25 over 9 yields the negative of the fraction 25 over 9, end fraction, times, open parenthesis, D minus T, close parenthesis, equals, 100 minus H. Subtracting 100 from both sides yields the negative of the fraction 25 over 9, end fraction, times, open parenthesis, D minus T, close parenthesis, minus 100, equals negative H. Multiplying both sides by negative 1 results in the formula the fraction 25 over 9, end fraction, times, open parenthesis, D minus T, close parenthesis, plus 100, equals H..

Choices B, C, and D are incorrect and may result from errors made when rewriting the given formula.

 

Question 298 298 of 479 selected Nonlinear Functions M

The number of bacteria in a liquid medium doubles every day. There are 44,000 bacteria in the liquid medium at the start of an observation. Which represents the number of bacteria, y , in the liquid medium t  days after the start of the observation? 

  1. y=12(44,000)t

  2. y=2(44,000)t

  3. y=44,000(12)t

  4. y=44,000(2)t

Show Answer Correct Answer: D

Choice D is correct. Since the number of bacteria doubles every day, the relationship between t and y can be represented by an exponential equation of the form y=a(b)t, where a is the number of bacteria at the start of the observation and the number of bacteria increases by a factor of b every day. It’s given that there are 44,000 bacteria at the start of the observation. Therefore, a = 44,000 . It’s also given that the number of bacteria doubles, or increases by a factor of 2 , every day. Therefore, b = 2 . Substituting 44,000 for a and 2 for b in the equation y=a(b)t yields y=44,000(2)t.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This equation represents a situation where the number of bacteria is decreasing by half, not doubling, every day.

Question 299 299 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

x-29=(x-a)(x-29)

Which of the following are solutions to the given equation, where a is a constant and a>30?

  1. a
  2. a + 1
  3. 29
  1. I and II only

  2. I and III only

  3. II and III only

  4. I, II, and III

Show Answer Correct Answer: C

Choice C is correct. Subtracting the expression (x-29) from both sides of the given equation yields 0=(x-a)(x-29)-(x-29), which can be rewritten as 0=(x-a)(x-29)+(-1)(x-29). Since the two terms on the right-hand side of this equation have a common factor of (x-29), it can be rewritten as 0=(x-29)(x-a+(-1)), or 0=(x-29)(x-a-1). Since x-a-1 is equivalent to x-(a+1), the equation 0=(x-29)(x-a-1) can be rewritten as 0=(x-29)(x-(a+1)). By the zero product property, it follows that x-29=0 or x-(a+1)=0. Adding 29 to both sides of the equation x-29=0 yields x = 29 . Adding a+1 to both sides of the equation x-(a+1)=0 yields x=a+1. Therefore, the two solutions to the given equation are 29 and a + 1 . Thus, only a + 1 and 29 , not a , are solutions to the given equation.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 300 300 of 479 selected Equivalent Expressions E

Which of the following expressions is equivalent to 2, a squared, times, open parenthesis, a, plus 3, close parenthesis  ?

  1. 5, a cubed

  2. 8, a to the fifth power

  3. 2, a cubed, plus 3

  4. 2, a cubed, plus 6, a squared

Show Answer Correct Answer: D

Choice D is correct. Expanding the given expression using the distributive property yields 2 a, squared times, a, plus 2 a, squared times, 3. Combining like terms yields 2 a, squared times, open parenthesis, a, to the first power, close parenthesis, plus, open parenthesis, 2 times 3, close parenthesis, times, open parenthesis, a, squared, close parenthesis, or 2 a, raised to the 2 plus 1 power, plus 6 a, squared, which is equivalent to 2 a, cubed, plus 6 a, squared.

Choices A and B are incorrect and may result from incorrectly combining like terms. Choice C is incorrect and may result from distributing 2 a, squared only to a, and not to 3, in the given expression.

 

Question 301 301 of 479 selected Equivalent Expressions H

Which of the following is equivalent to parenthesis, a, plus the fraction b over 2, close parenthesis, squared ?

  1. a, squared, plus the fraction b squared over 2

  2. a, squared, plus the fraction b squared over 4

  3. a, squared, plus the fraction a, times b, over 2, end fraction, plus the fraction b squared over 2

  4. a, squared, plus a, times b, plus the fraction b squared over 4

Show Answer Correct Answer: D

Choice D is correct. The expression open parenthesis, a, plus the fraction b over 2, close parenthesis, squared can be rewritten as open parenthesis, a, plus the fraction b over 2, close parenthesis, times, open parenthesis, a, plus the fraction b over 2, close parenthesis. Using the distributive property, the expression yields open parenthesis, a, plus the fraction b over 2, close parenthesis, times, open parenthesis, a, plus the fraction b over 2, close parenthesis, equals, a, squared, plus the fraction with numerator a, b and denominator 2, end fraction, plus the fraction with numerator a, b and denominator 2, end fraction, plus the fraction b squared, over 4. Combining like terms gives a, squared, plus a, b, plus the fraction b squared over 4.

Choices A, B, and C are incorrect and may result from errors using the distributive property on the given expression or combining like terms.

 

Question 302 302 of 479 selected Nonlinear Functions M

f(x)= x 2 - 18 x - 360

If the given function f is graphed in the xy-plane, where y=f(x), what is an x-intercept of the graph?

  1. (-12,0)

  2. (-30,0)

  3. (-360,0)

  4. (12,0)

Show Answer Correct Answer: A

Choice A is correct. It's given that y=f(x). The x-intercepts of a graph in the xy-plane are the points where y = 0 . Thus, for an x-intercept of the graph of function f , 0=f(x). Substituting 0 for f(x) in the equation f(x)=x2-18x-360 yields 0 = x 2 - 18 x - 360 . Factoring the right-hand side of this equation yields 0=(x+12)(x-30). By the zero product property, x + 12 = 0 and x - 30 = 0 . Subtracting 12 from both sides of the equation x + 12 = 0 yields x = -12 . Adding 30 to both sides of the equation x - 30 = 0 yields x = 30 . Therefore, the x-intercepts of the graph of y=f(x) are (-12,0) and (30,0). Of these two x-intercepts, only (-12,0) is given as a choice.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 303 303 of 479 selected Nonlinear Functions E
The figure presents the graph of a curve. The x axis is labeled “Elevation, in meters,” and the numbers 0 through 3,000, in increments of 500, are indicated. There are vertical gridlines drawn in between. The y axis is labeled “Measure of plant diversity,” and the numbers 0 through 14,000, in increments of 2,000, are indicated. The curve begins on the vertical axis at a point slightly above 10,000. The curve moves upward and to the right until it reaches its maximum at the point with coordinates 1,250 comma 13,000. Then the curve moves downward and to the right until it ends at 3,500 on the horizontal axis.

The quadratic function graphed above models a particular measure of plant diversity as a function of the elevation in a region of Switzerland. According to the model, which of the following is closest to the elevation, in meters, at which plant diversity is greatest?

  1. 13,500

  2. 3,000

  3. 1,250

  4. 250

Show Answer Correct Answer: C

Choice C is correct. Each point with coordinates x comma y on the graph represents the elevation x, in meters, and the corresponding measure of plant diversity y in a region of Switzerland. Therefore, the point on the graph with the greatest y-coordinate represents the location that has the greatest measure of plant diversity in the region. The greatest y-coordinate of any point on the graph is approximately 13,500. The x-coordinate of that point is approximately 1,250. Therefore, the closest elevation at which the plant diversity is the greatest is 1,250 meters.

Choice A is incorrect. This value is closest to the greatest y-coordinate of any point on the graph and therefore represents the greatest measure of plant diversity, not the elevation where the greatest measure of plant diversity occurs. Choice B is incorrect. At an elevation of 3,000 meters the measure of plant diversity is approximately 4,000. Because there are points on the graph with greater y-coordinates, 4,000 can’t be the greatest measure of plant diversity, and 3,000 meters isn’t the elevation at which the greatest measure of plant diversity occurs. Choice D is incorrect. At an elevation of 250 meters, the measure of plant diversity is approximately 11,000. Because there are points on the graph with greater y-coordinates, 11,000 can’t be the greatest measure of plant diversity and 250 meters isn’t the elevation at which the greatest measure of plant diversity occurs.

 

Question 304 304 of 479 selected Nonlinear Functions H

Square P has a side length of x inches. Square Q has a perimeter that is 176 inches greater than the perimeter of square P. The function f gives the area of square Q, in square inches. Which of the following defines f ?

  1. f(x)=(x+44)2

  2. f(x)=(x+176)2

  3. f(x)=(176x+44)2

  4. f(x)=(176x+176)2

Show Answer Correct Answer: A

Choice A is correct. Let x represent the side length, in inches, of square P. It follows that the perimeter of square P is 4x inches. It's given that square Q has a perimeter that is 176 inches greater than the perimeter of square P. Thus, the perimeter of square Q is 176 inches greater than 4x inches, or 4x+176 inches. Since the perimeter of a square is 4 times the side length of the square, each side length of Q is 4x+1764, or x+44 inches. Since the area of a square is calculated by multiplying the length of two sides, the area of square Q is (x+44)(x+44), or (x+44)2 square inches. It follows that function f is defined by f(x)=(x+44)2

Choice B is incorrect. This function represents a square with side lengths (x+176) inches.

Choice C is incorrect. This function represents a square with side lengths (176x+44) inches.

Choice D is incorrect. This function represents a square with side lengths (176x+176) inches.

Question 305 305 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

y = 2 x 2 - 21 x + 64

y = 3 x + a

In the given system of equations, a is a constant. The graphs of the equations in the given system intersect at exactly one point, (x,y), in the xy-plane. What is the value of x ?

  1. -8

  2. -6

  3. 6

  4. 8

Show Answer Correct Answer: C

Choice C is correct. It's given that the graphs of the equations in the given system intersect at exactly one point, (x,y), in the xy-plane. Therefore, (x,y) is the only solution to the given system of equations. The given system of equations can be solved by subtracting the second equation, y=3x+a, from the first equation, y=2x2-21x+64. This yields y-y=(2x2-21x+64)-(3x+a), or 0=2x2-24x+64-a. Since the given system has only one solution, this equation has only one solution. A quadratic equation in the form rx2+sx+t=0, where r , s , and t are constants, has one solution if and only if the discriminant, s2-4rt, is equal to zero. Substituting 2 for r , -24 for s , and - a + 64 for t in the expression s2-4rt yields (-24)2-(4)(2)(64-a). Setting this expression equal to zero yields (-24)2-(4)(2)(64-a)=0, or 8 a + 64 = 0 . Subtracting 64 from both sides of this equation yields 8a=-64. Dividing both sides of this equation by 8 yields a=-8. Substituting -8 for a in the equation 0=2x2-24x+64-a yields 0=2x2-24x+64+8, or 0=2x2-24x+72. Factoring 2 from the right-hand side of this equation yields 0=2(x2-12x+36). Dividing both sides of this equation by 2 yields 0=x2-12x+36, which is equivalent to 0=(x-6)(x-6), or 0=(x-6)2. Taking the square root of both sides of this equation yields 0 = x - 6 . Adding 6 to both sides of this equation yields x = 6

Choice A is incorrect. This is the value of a , not x .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 306 306 of 479 selected Nonlinear Functions H

The surface area of a cube is 6 times, open parenthesis, the fraction a, over 4, close parenthesis, squared, where a is a positive constant. Which of the following gives the perimeter of one face of the cube?

  1. the fraction a, over 4

  2. a

  3. 4 a

  4. 6 a

Show Answer Correct Answer: B

Choice B is correct. A cube has 6 faces of equal area, so if the total surface area of a cube is 6 times, open parenthesis, a, over 4, close parenthesis, squared, then the area of one face is open parenthesis, a, over 4, close parenthesis, squared. Likewise, the area of one face of a cube is the square of one of its edges; therefore, if the area of one face is open parenthesis, a, over 4, close parenthesis, squared, then the length of one edge of the cube is a, over 4. Since the perimeter of one face of a cube is four times the length of one edge, the perimeter is 4 times the fraction a, over 4, equals a.
Choice A is incorrect because if the perimeter of one face of the cube is a, over 4, then the total surface area of the cube is 6 times, open parenthesis, the fraction with numerator a, over 4, and denominator 4, close parenthesis, squared, equals, 6 times, open parenthesis, a, over 16, close parenthesis, squared, which is not 6 times, open parenthesis, a, over 4, close parenthesis, squared. Choice C is incorrect because if the perimeter of one face of the cube is 4a, then the total surface area of the cube is 6 times, open parenthesis, the fraction 4 a, over 4, close parenthesis, squared, equals 6 a, squared, which is not 6 times, open parenthesis, a, over 4, close parenthesis, squared. Choice D is incorrect because if the perimeter of one face of the cube is 6a, then the total surface area of the cube is 6 times, open parenthesis, the fraction 6 a, over 4, close parenthesis, squared, equals, 6 times, open parenthesis, the fraction 3 a, over 2, close parenthesis, squared, which is not 6 times, open parenthesis, a, over 4, close parenthesis, squared.

Question 307 307 of 479 selected Nonlinear Functions E

  • For the first curve:
    • Moving from left to right, the curve is in quadrant 3.
    • As x decreases, the curve approaches the line y equals negative 1.
    • As x increases, the curve approaches the line x equals negative 1.
  • For the second curve:
    • Moving from left to right, the curve passes from quadrant 2 to quadrant 1 to quadrant 4.
    • As x decreases, the curve approaches the line x equals negative 1.
    • As x increases, the curve approaches the line y equals negative 1.
  • The 2 curves pass through the following points:
    • (negative 6 comma negative nine fifths)
    • (0 comma 3)
    • (4 comma negative one fifth)

What is the y-coordinate of the y-intercept of the graph shown?

Show Answer Correct Answer: 3

The correct answer is 3 . A y-intercept of a graph in the xy-plane is a point (x,y) on the graph where x = 0 . For the graph shown, at x = 0 , the corresponding value of y is 3 . Therefore, the y-coordinate of the y-intercept of the graph shown is 3 .

Question 308 308 of 479 selected Nonlinear Functions M

The function f is defined by f(x)=7x3. In the xy-plane, the graph of y=g(x) is the result of shifting the graph of y=f(x) down 2 units. Which equation defines function g ?

  1. g(x)=72x3

  2. g(x)=7x32

  3. g(x)=7x3+2

  4. g(x)=7x3-2

Show Answer Correct Answer: D

Choice D is correct. If the graph of y=g(x) is the result of shifting the graph of y=f(x) down k units in the xy-plane, the function g can be defined by an equation of the form g(x)=f(x)-k. It’s given that f(x)=7x3 and the graph of y=g(x) is the result of shifting the graph of y=f(x) down 2 units. Substituting 7x3 for f(x) and 2 for k in the equation g(x)=f(x)-k yields g(x)=7x3-2

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect. This equation defines a function g for which the graph of y=g(x) is the result of shifting the graph of y=f(x) up, not down, 2 units.

Question 309 309 of 479 selected Nonlinear Functions M
x y
1 11
2 19
3 a

The table shows three values of x and their corresponding values of y for the equation y=4(2)x+3. In the table, a is a constant. What is the value of a ?

  1. 67

  2. 35

  3. 32

  4. 27

Show Answer Correct Answer: B

Choice B is correct. It's given that the table shows three values of x and their corresponding values of y for the equation y=4(2)x+3. It's also given that when x = 3 the corresponding value of y is a , and a is a constant. Substituting 3 for x and a for y in the given equation yields a=4(2)3+3, or a = 35 . Therefore, the value of a is 35 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 310 310 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

The equation 12t+b=c relates the variables t , b , and c . Which of the following correctly expresses the value of c-b in terms of t ?

  1. t 12

  2. t

  3. t+112

  4. 12 t

Show Answer Correct Answer: D

Choice D is correct. Subtracting b from each side of the given equation yields 12t=c-b. Therefore, the expression 12 t correctly expresses the value of c-b in terms of t

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 311 311 of 479 selected Nonlinear Functions E

  • Moving from left to right:
    • The curve passes from quadrant 2 to quadrant 1.
    • In quadrant 2, the curve trends up sharply to the point (0 comma 3).
    • In quadrant 1, the curve trends up sharply.
  • The curve passes through the following points:
    • approximately (negative 1 comma 0.2)
    • (0 comma 3)

The graph of the exponential function f is shown, where y=f(x). The y-intercept of the graph is (0,y). What is the value of y ?

Show Answer Correct Answer: 3

The correct answer is 3 . For the graph of the exponential function f shown, where y=f(x), it's given that the y-intercept of the graph is (0,y). The graph intersects the y-axis at the point (0,3). Therefore, the value of y is 3 .

Question 312 312 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

Equation 1: x squared equals, 6 x plus y. 
Equation 2: y equals, negative 6 x, plus 36.

A solution to the given system of equations is the ordered pair x comma y. Which of the following is a possible value of xy ?

  1. 0

  2. 6

  3. 12

  4. 36

Show Answer Correct Answer: A

Choice A is correct. Solutions to the given system of equations are ordered pairs x comma y that satisfy both equations in the system. Adding the left-hand and right-hand sides of the equations in the system yields x squared, plus y, equals, 6 x plus negative 6 x, plus y, plus 36, or x squared, plus y, equals, y plus 36. Subtracting y from both sides of this equation yields x squared equals 36. Taking the square root of both sides of this equation yields x equals 6 and x equals negative 6. Therefore, there are two solutions to this system of equations, one with an x-coordinate of 6 and the other with an x-coordinate of negative 6. Substituting 6 for x in the second equation yields y equals, negative 6 times 6, plus 36, or y equals 0; therefore, one solution is the ordered pair 6 comma 0. Similarly, substituting negative 6 for x in the second equation yields y equals, negative 6 times negative 6, plus 36, or y equals 72; therefore, the other solution is the ordered pair negative 6 comma 72. It follows then that if the ordered pair x comma y is a solution to the system, then possible values of x y   are 6 times 0, equals 0 and negative 6 times 72, equals negative 432. Only 0 is among the given choices.

Choice B is incorrect. This is the x-coordinate of one of the solutions, the ordered pair 6 comma 0. Choice C is incorrect and may result from conceptual or computational errors. Choice D is incorrect. This is the square of the x-coordinate of one of the solutions, the ordered pair 6 comma 0.

Question 313 313 of 479 selected Nonlinear Functions M

According to Moore’s law, the number of transistors included on microprocessors doubles every 2 years. In 1985, a microprocessor was introduced that had 275,000 transistors. Based on this information, in which of the following years does Moore’s law estimate the number of transistors to reach 1.1 million?

  1. 1987

  2. 1989

  3. 1991

  4. 1994

Show Answer Correct Answer: B

Choice B is correct. Let x be the number of years after 1985. It follows that x over 2 represents the number of 2-year periods that will occur within an x-year period. According to Moore’s law, every 2 years, the number of transistors included on microprocessors is estimated to double. Therefore, x years after 1985, the number of transistors will double x over 2 times. Since the number of transistors included on a microprocessor was 275,000, or .275 million, in 1985, the estimated number of transistors, in millions, included x years after 1985 can be modeled as 0 point 2 7 5 times 2 raised to the x over 2 power. The year in which the number of transistors is estimated to be 1.1 million is represented by the value of x when 1 point 1 equals, 0 point 2 7 5, times 2 raised to the x over 2 power. Dividing both sides of this equation by .275 yields 4 equals, 2 raised to the x over 2 power, which can be rewritten as 2 squared equals, 2 raised to the x over 2 power. Since the exponential equation has equal bases on each side, it follows that the exponents must also be equal: 2 equals x over 2. Multiplying both sides of the equation 2 equals x over 2 by 2 yields x equals 4. Therefore, according to Moore’s law, 4 years after 1985, or in 1989, the number of transistors included on microprocessors is estimated to reach 1.1 million.

Alternate approach: According to Moore’s law, 2 years after 1985 (in 1987), the number of transistors included on a microprocessor is estimated to be 2 times 275,000, or 550,000, and 2 years after 1987 (in 1989), the number of transistors included on microprocessors is estimated to be 2 times 550,000, or 1,100,000. Therefore, the year that Moore’s law estimates the number of transistors on microprocessors to reach 1.1 million is 1989.

Choices A, C, and D are incorrect. According to Moore’s law, the number of transistors included on microprocessors is estimated to reach 550,000 in 1987, 2.2 million in 1991, and about 6.2 million in 1994.

Question 314 314 of 479 selected Nonlinear Functions M

The function f is defined by f(x)=270(0.1)x. What is the value of f(0)?

  1. 0

  2. 1

  3. 27

  4. 270

Show Answer Correct Answer: D

Choice D is correct. The value of f(0) is the value of f(x) when x = 0 . Substituting 0 for x in the given function yields f(0)=270(0.1)0, or f(0)=270(1), which is equivalent to f(0)=270. Therefore, the value of f(0) is 270 .

Choice A is incorrect. This is the value of x , not f(x).

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the value of f(1), not f(0).

Question 315 315 of 479 selected Nonlinear Functions E

The function g is defined by g(x)=x2+9. For which value of x is g(x)=25?

  1. 4

  2. 5

  3. 9

  4. 13

Show Answer Correct Answer: A

Choice A is correct. It's given that g(x)=x2+9. Substituting 25 for g(x) in this equation yields 25=x2+9. Subtracting 9 from both sides of this equation yields 16=x2. Taking the square root of each side of this equation yields x=±4. It follows that g(x)=25 when the value of x is 4 or -4 . Only 4 is listed among the choices.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 316 316 of 479 selected Equivalent Expressions E

Which expression is equivalent to (x)114, where x>0?

  1. 114·x

  2. x14

  3. 14·x

  4. (x)14

Show Answer Correct Answer: B

Choice B is correct. An expression in the form x1k, where x>0 and k>0, is equivalent to xk. It follows that x114, where x>0, is equivalent to x14.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 317 317 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

x squared plus x, minus 12, equals 0

If a is a solution of the equation above and a is greater than 0, what is the value of a ?

Show Answer

The correct answer is 3. The solution to the given equation can be found by factoring the quadratic expression. The factors can be determined by finding two numbers with a sum of 1 and a product of negative 12. The two numbers that meet these constraints are 4 and negative 3. Therefore, the given equation can be rewritten as open parenthesis, x plus 4, close parenthesis, times, open parenthesis, x minus 3, close parenthesis, equals 0. It follows that the solutions to the equation are x equals negative 4 or x equals 3. Since it is given that a, is greater than 0, a must equal 3.

Question 318 318 of 479 selected Equivalent Expressions E

Which expression is equivalent to 5 x 2 - 50 x y 2 ?

  1. 5x(x-10y2)

  2. 5x(x-50y2)

  3. 5x2(10xy2)

  4. 5x2(50xy2)

Show Answer Correct Answer: A

Choice A is correct. Since each term of the given expression has a factor of 5 x , it can be rewritten as 5x(x)-5x(10y2), or 5x(x-10y2).

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 319 319 of 479 selected Nonlinear Functions E

f of x equals, x squared, plus 4

The function f is defined as shown. Which of the following graphs in the xy-plane could be the graph of y equals, f of x ?

  1. ​​​​​​​The answer choice presents the graph of a curve in the x y plane, with the origin labeled O. The numbers negative 8, negative 4, 4, and 8 are indicated on each axis. There are horizontal gridlines at the numbers negative 8 through 8, in increments of 2, on the y axis, and there are vertical gridlines at the numbers negative 8 through 8, in increments of 2, on the x axis. The figure presents an upward opening parabola, whose vertex lies at negative 4 on the y axis. The parabola crosses the x axis at negative 2 and 2.

     

  2. ​​​​​​​​​​​​​The answer choice presents the graph of a curve in the x y plane, with the origin labeled O. The numbers negative 8, negative 4, 4, and 8 are indicated on each axis. There are horizontal gridlines at the numbers negative 8 through 8, in increments of 2, on the y axis, and there are vertical gridlines at the numbers negative 8 through 8, in increments of 2, on the x axis. The figure presents an upward opening parabola, whose vertex lies at negative 4 on the x axis. The parabola passes through the point with coordinates negative 6 comma 4, and the point with coordinates negative 2 comma 4.

     

  3. ​​​​​​​The answer choice presents the graph of a curve in the x y plane, with the origin labeled O. The numbers negative 8, negative 4, 4, and 8 are indicated on each axis. There are horizontal gridlines at the numbers negative 8 through 8, in increments of 2, on the y axis, and there are vertical gridlines at the numbers negative 8 through 8, in increments of 2, on the x axis. The figure presents an upward opening parabola, whose vertex lies at 4 on the x axis. The parabola passes through the point with coordinates 2 comma 4, and the point with coordinates 6 comma 4.

     

  4. ​​​​​​​The answer choice presents the graph of a curve in the x y plane, with the origin labeled O. The numbers negative 8, negative 4, 4, and 8 are indicated on each axis. There are horizontal gridlines at the numbers negative 8 through 8, in increments of 2, on the y axis, and there are vertical gridlines at the numbers negative 8 through 8, in increments of 2, on the x axis. The figure presents an upward opening parabola, whose vertex lies at 4 on the y axis. The parabola passes through the point with coordinates negative 2 comma 8, and the point with coordinates 2 comma 8.

     

Show Answer Correct Answer: D

Choice D is correct. For the quadratic function f of x equals, x squared, plus 4, the vertex of the graph is the point with coordinates 0 comma 4, and because the x squared term is positive, the vertex is the minimum of the function. Choice D is the only option that meets these conditions.

Choices A, B, and C are incorrect. The vertex of each of these graphs doesn’t correspond to the minimum of the given function.

 

Question 320 320 of 479 selected Nonlinear Functions M

The function f(x)=206(1.034)x models the value, in dollars, of a certain bank account by the end of each year from 1957 through 1972 , where x is the number of years after 1957 . Which of the following is the best interpretation of “f(5) is approximately equal to 243 ” in this context?

  1. The value of the bank account is estimated to be approximately 5 dollars greater in 1962 than in 1957 .

  2. The value of the bank account is estimated to be approximately 243 dollars in 1962 .

  3. The value, in dollars, of the bank account is estimated to be approximately 5 times greater in 1962 than in 1957 .

  4. The value of the bank account is estimated to increase by approximately 243 dollars every 5 years between 1957 and 1972 .

Show Answer Correct Answer: B

Choice B is correct. It’s given that the function f(x)=206(1.034)x models the value, in dollars, of a certain bank account by the end of each year from 19 57 through 19 72 , where x is the number of years after 19 57 . It follows that f(x) represents the estimated value, in dollars, of the bank account x years after 19 57 . Since the value of f(5) is the value of f(x) when x = 5 , it follows that “f(5) is approximately equal to 243 ” means that f(x) is approximately equal to 243 when x = 5 . In the given context, this means that the value of the bank account is estimated to be approximately 243 dollars 5 years after 19 57 . Therefore, the best interpretation of the statement “f(5) is approximately equal to 243 ” in this context is the value of the bank account is estimated to be approximately 243 dollars in 19 62 .

Choice A is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 321 321 of 479 selected Equivalent Expressions M

Which expression is equivalent to (x2+11)2+(x-5)(x+5)?

  1. x 4 + 23 x 2 - 14

  2. x 4 + 23 x 2 + 96

  3. x 4 + 12 x 2 + 121

  4. x 4 + x 2 + 146

Show Answer Correct Answer: B

Choice B is correct. The expression (x2+11)2 can be written as (x2+11)(x2+11), which is equivalent to x2(x2+11)+11(x2+11). Distributing x2 and 11 to (x2+11) yields x4+11x2+11x2+121, or x4+22x2+121. The expression (x-5)(x+5) is equivalent to (x-5)x+(x-5)5. Distributing x and 5 to (x-5) yields x2-5x+5x-25, or x2-25. Therefore, the expression (x2+11)2+(x-5)(x+5) is equivalent to (x4+22x2+121)+(x2-25), or x4+22x2+121+x2-25. Combining like terms in this expression yields x4+23x2+96.

Choice A is incorrect. Equivalent expressions must be equivalent for any value of x . Substituting 0 for x in this expression yields -14 , whereas substituting 0 for x in the given expression yields 96 .

Choice C is incorrect. Equivalent expressions must be equivalent for any value of x . Substituting 0 for x in this expression yields 121 , whereas substituting 0 for x in the given expression yields 96 .

Choice D is incorrect. Equivalent expressions must be equivalent for any value of x . Substituting 0 for x in this expression yields 146 , whereas substituting 0 for x in the given expression yields 96 .

Question 322 322 of 479 selected Nonlinear Functions M

  • Moving from left to right:
    • The curve passes through quadrant 1.
    • The curve trends up sharply.
  • The curve passes through the following points:
    • (0 comma 6)
    • (1 comma 9.0)

The graph gives the estimated population y , in thousands, of a town x years since 2003 , where 0x5. Which of the following best describes the increase in the estimated population from x=0 to x=1?

  1. The estimated population at x=1 is 0.5 times the estimated population at x=0.

  2. The estimated population at x=1 is 1.5 times the estimated population at x=0.

  3. The estimated population at x=1 is 2.5 times the estimated population at x=0.

  4. The estimated population at x=1 is 3.5 times the estimated population at x=0.

Show Answer Correct Answer: B

Choice B is correct. On the graph shown, the y-axis represents estimated population, in thousands. The graph shows that when x = 0 , the y-coordinate is 6 . Therefore, the estimated population at x = 0 is 6 thousand. The graph also shows that when x = 1 , the y-coordinate is 9 . Therefore, the estimated population at x = 1 is 9 thousand. Dividing 9 thousand by 6 thousand yields 1.5 ; therefore, 9 thousand is 1.5 times 6 thousand. It follows that the estimated population at x = 1 is 1.5 times the estimated population at x = 0 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 323 323 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

In the xy-plane, the graph of y equals 3 x squared minus 14 x intersects the graph of y equals x at the points with coordinates zero comma zero and a, comma a. What is the value of a ?

Show Answer

The correct answer is 5. The intersection points of the graphs of y equals, 3 x squared, minus 14 x and y equals x can be found by solving the system consisting of these two equations. To solve the system, substitute x for y in the first equation. This gives x equals, 3 x squared, minus 14 x. Subtracting x from both sides of the equation gives 0 equals, 3 x squared, minus 15 x. Factoring 3 x out of each term on the left-hand side of the equation gives 0 equals, 3 x times, open parenthesis, x minus 5, close parenthesis. Therefore, the possible values for x are 0 and 5. Since y equals x, the two intersection points are the points with coordinates 0 comma 0 and 5 comma 5. Therefore, a, equals 5.

Question 324 324 of 479 selected Nonlinear Functions M

h of t equals, negative 16 t squared, plus 110 t, plus 72

The function above models the height h, in feet, of an object above ground t seconds after being launched straight up in the air. What does the number 72 represent in the function?

  1. The initial height, in feet, of the object

  2. The maximum height, in feet, of the object

  3. The initial speed, in feet per second, of the object

  4. The maximum speed, in feet per second, of the object

Show Answer Correct Answer: A

Choice A is correct. The variable t represents the seconds after the object is launched. Since h of 0 equals 72, this means that the height, in feet, at 0 seconds, or the initial height, is 72 feet.

Choices B, C, and D are incorrect and may be the result of misinterpreting the function in context.

 

Question 325 325 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

x 2 - 5 x - 24 = 0

What is the sum of the solutions to the given equation?

Show Answer Correct Answer: 5

The correct answer is 5. The given quadratic equation can be rewritten in factored form as (x-8)(x+3)=0. Based on the zero product property, it follows that x-8=0 or x+3=0. Adding 8 to both sides of the equation x-8=0 yields x=8. Subtracting 3 from both sides of the equation x+3=0 yields x=-3. Therefore, the solutions to the given equation are 8 and -3. It follows that the sum of the solutions to the given equation is 8+(-3), or 5.

Question 326 326 of 479 selected Nonlinear Functions M

A model predicts that the population of Bergen was 15,000 in 2005. The model also predicts that each year for the next 5 years, the population p  increased by 4% of the previous year's population. Which equation best represents this model, where x  is the number of years after 2005, for x5?

  1. p=0.96(15,000)x

  2. p=1.04(15,000)x

  3. p=15,000(0.96)x

  4. p=15,000(1.04)x

Show Answer Correct Answer: D

Choice D is correct. It's given that a model predicts the population of Bergen in 2005 was 15,000 . The model also predicts that each year for the next 5 years, the population increased by 4% of the previous year's population. The predicted population in one of these years can be found by multiplying the predicted population from the previous year by 1.04 . Since the predicted population in 2005 was 15,000 , the predicted population 1 year later is 15,000(1.04). The predicted population 2 years later is this value times 1.04 , which is 15,000(1.04)(1.04), or 15,000(1.04)2. The predicted population 3 years later is this value times 1.04 , or 15,000(1.04)3. More generally, the predicted population, p , x years after 2005 is represented by the equation p=15,000(1.04)x

Choice A is incorrect. Substituting 0 for x in this equation indicates the predicted population in 2005 was 0.96 rather than 15,000 .

Choice B is incorrect. Substituting 0 for x in this equation indicates the predicted population in 2005 was 1.04 rather than 15,000 .

Choice C is incorrect. This equation indicates the predicted population is decreasing, rather than increasing, by 4% each year.

Question 327 327 of 479 selected Nonlinear Functions M

N of d, equals, 115 times, open parenthesis, 0.90, close parenthesis, raised to the d power.

The function N defined above can be used to model the number of species of brachiopods at various ocean depths d, where d is in hundreds of meters. Which of the following does the model predict?

  1. For every increase in depth by 1 meter, the number of brachiopod species decreases by 115.

  2. For every increase in depth by 1 meter, the number of brachiopod species decreases by 10%.

  3. For every increase in depth by 100 meters, the number of brachiopod species decreases by 115.

  4. For every increase in depth by 100 meters, the number of brachiopod species decreases by 10%.

Show Answer Correct Answer: D

Choice D is correct. The function N is exponential, so it follows that N of d changes by a fixed percentage for each increase in d by 1. Since d is measured in hundreds of meters, it also follows that the number of brachiopod species changes by a fixed percentage for each increase in ocean depth by 100 meters. Since the base of the exponent in the model is 0.90, which is less than 1, the number of brachiopod species decreases as the ocean depth increases. Specifically, the number of brachiopod species at a depth of d plus 100 meters is 90% of the number of brachiopod species at a depth of d meters. This means that for each increase in ocean depth by 100 meters, the number of brachiopod species decreases by 10%.

Choices A and C are incorrect. These describe situations where the number of brachiopod species are decreasing linearly rather than exponentially. Choice B is incorrect and results from interpreting the decrease in the number of brachiopod species as 10% for every 1-meter increase in ocean depth rather than for every 100-meter increase in ocean depth.

 

Question 328 328 of 479 selected Nonlinear Functions H

y=576(2x+2)

The graph of the given equation in the xy-plane has a y-intercept of (r,s). Which of the following equivalent equations displays the value of s as a constant, a coefficient, or the base?

  1. y=331,776(x+1)

  2. y=24(4x+4)

  3. y=124(24)(4x+5)

  4. y=1576(576)(2x+3)

Show Answer Correct Answer: A

Choice A is correct. The y-intercept of a graph in the xy-plane is the point where x = 0 . Substituting 0 for x in the given equation, y=576(2x+2), yields y=576(2(0)+2), which is equivalent to y=5762, or y = 331,776 . Therefore, the graph of the given equation in the xy-plane has a y-intercept of (0,331,776). It follows that r = 0 and s = 331,776 . Thus, the equivalent equation y=331,776(x+1) displays the value of s as the base.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 329 329 of 479 selected Equivalent Expressions H

Which expression is equivalent to y+12x-8+y(x-8)x2y-8xy?

  1. x y + y + 4 x 3 y - 16 x 2 y + 64 x y

  2. x y + 9 y + 12 x 2 y - 8 x y + x - 8

  3. x y 2 + 13 x y - 8 y x 2 y - 8 x y

  4. x y 2 + 13 x y - 8 y x 3 y - 16 x 2 y + 64 x y

Show Answer Correct Answer: C

Choice C is correct. Factoring the denominator in the second term of the given expression gives y+12x-8+y(x-8)xy(x-8). This expression can be rewritten with common denominators by multiplying the first term by xyxy, giving xy(y+12)xy(x-8)+y(x-8)xy(x-8). Adding these two terms yields xy(y+12)+y(x-8)xy(x-8). Using the distributive property to rewrite this expression gives xy2+12xy+xy-8yx2y-8xy. Combining the like terms in the numerator of this expression gives xy2+13xy-8yx2y-8xy.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 330 330 of 479 selected Equivalent Expressions H

One of the factors of 2 x 3 + 42 x 2 + 208 x is x + b , where b  is a positive constant. What is the smallest possible value of b ?

Show Answer Correct Answer: 8

The correct answer is 8 . Since each term of the given expression, 2x3+42x2+208x, has a factor of 2 x , the expression can be rewritten as 2x(x2)+2x(21x)+2x(104), or 2x(x2+21x+104). Since the values 8 and 13 have a sum of 21 and a product of 104 , the expression x2+21x+104 can be factored as (x+8)(x+13). Therefore, the given expression can be factored as 2x(x+8)(x+13). It follows that the factors of the given expression are 2 , x , x + 8 , and x + 13 . Of these factors, only x + 8 and x + 13 are of the form x + b , where b is a positive constant. Therefore, the possible values of b are 8 and 13 . Thus, the smallest possible value of b is 8 .

Question 331 331 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

x + 7 = 10

(x+7)2=y

Which ordered pair (x,y) is a solution to the given system of equations?

  1. (3,100)

  2. (3,3)

  3. (3,10)

  4. (3,70)

Show Answer Correct Answer: A

Choice A is correct. The solution to a system of equations is the ordered pair (x,y) that satisfies all equations in the system. It's given by the first equation in the system that x+7=10. Substituting 10 for x + 7 into the second equation yields 102=y, or y=100. The x-coordinate of the solution to the system of equations can be found by subtracting 7 from both sides of the equation x+7=10, which yields x=3. Therefore, the ordered pair (3,100) is a solution to the given system of equations.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 332 332 of 479 selected Nonlinear Functions H

The function h is defined by h(x)=ax+b, where a and b are positive constants. The graph of y=h(x) in the x y -plane passes through the points (0 , 10 ) and (-2 , 325 36 ). What is the value of a b ?

  1. 1 4

  2. 1 2

  3. 54

  4. 60

Show Answer Correct Answer: C

Choice C is correct. It’s given that the function h is defined by h(x)=ax+b and that the graph of y=h(x) in the xy-plane passes through the points (0,10) and (-2, 32536). Substituting 0 for x and 10 for h(x) in the equation h(x)=ax+b yields 10=a0+b, or 10=1+b. Subtracting 1 from both sides of this equation yields 9=b. Substituting -2 for x and 32536 for h(x) in the equation h(x)=ax+9 yields 32536=a-2+9. Subtracting 9 from both sides of this equation yields 136=a-2, which can be rewritten as a2=36. Taking the square root of both sides of this equation yields a=6 and a=-6, but because it’s given that a is a positive constant, a must equal 6 . Because the value of a is 6 and the value of b is 9 , the value of a b is (6)(9), or 54 .


Choice A is incorrect and may result from finding the value of a-2b rather than the value of ab.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from correctly finding the value of a as 6 , but multiplying it by the y-value in the first ordered pair rather than by the value of b .

Question 333 333 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

T equals, 0 point 0 1 times, open parenthesis, P minus 40,000, close parenthesis

In a city, the property tax T, in dollars, is calculated using the formula above, where P is the value of the property, in dollars. Which of the following expresses the value of the property in terms of the property tax?

  1. P equals 100 T, minus 400

  2. P equals 100 T, plus 400

  3. P equals 100 T, minus 40,000

  4. P equals 100 T, plus 40,000

Show Answer Correct Answer: D

Choice D is correct. To express the value of the property in terms of the property tax, the given equation must be solved for P. Multiplying both sides of the equation by 100 gives 100 T equals, P minus 40,000. Adding 40,000 to both sides of the equation gives 100 T plus 40,000, equals P. Therefore, P equals, 100 T plus 40,000.

Choice A is incorrect and may result from multiplying 40,000 by 0.01, then subtracting 400 from, instead of adding 400 to, the left-hand side of the equation. Choice B is incorrect and may result from multiplying 40,000 by 0.01. Choice C is incorrect and may result from subtracting instead of adding 40,000 from the left-hand side of the equation.

 

Question 334 334 of 479 selected Nonlinear Functions H

The function f is defined by f(x)=ax+b, where a and b are constants. In the xy-plane, the graph of y=f(x) passes through the point (-24,0), and f(24)<0. Which of the following must be true?

  1. f(0)=24

  2. f(0)=-24

  3. a>b

  4. a<b

Show Answer Correct Answer: D

Choice D is correct. It's given that f(24)<0. Substituting 24 for f(x) in the equation f(x)=ax+b yields f(24)=a24+b. Therefore, a24+b<0. Since 24+b can't be negative, it follows that a<0. It's also given that the graph of y=f(x) passes through the point (-24,0). It follows that when x = -24 , f(x)=0. Substituting -24 for x and 0 for f(x) in the equation f(x)=ax+b yields 0=a-24+b. By the zero product property, either a = 0 or -24+b=0. Since a<0, it follows that 24+b=0. Squaring both sides of this equation yields -24+b=0. Adding 24 to both sides of this equation yields b = 24 . Since a<0 and b is 24 , it follows that a<b must be true.

Choice A is incorrect. The value of f(0) is a b , which must be negative.

Choice B is incorrect. The value of f(0) is a b , which could be -24 , but doesn't have to be.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 335 335 of 479 selected Nonlinear Functions H
x y
21 -8
23 8
25 -8

The table shows three values of x and their corresponding values of y , where y=f(x)+4 and f is a quadratic function. What is the y-coordinate of the y-intercept of the graph of y=f(x) in the xy-plane?

Show Answer Correct Answer: -2112

The correct answer is -2,112. It's given that f is a quadratic function. It follows that f can be defined by an equation of the form f(x)=a(x-h)2+k, where a , h , and k are constants. It's also given that the table shows three values of x and their corresponding values of y , where y=f(x)+4. Substituting a(x-h)2+k for f(x) in this equation yields y=a(x-h)2+k+4. This equation represents a quadratic relationship between x and y , where k + 4  is either the maximum or the minimum value of y , which occurs when x = h . For quadratic relationships between x and y , the maximum or minimum value of y occurs at the value of x halfway between any two values of x that have the same corresponding value of y . The table shows that x-values of 21 and 25 correspond to the same y-value, -8. Since 23 is halfway between 21 and 25 , the maximum or minimum value of y occurs at an x-value of 23 . The table shows that when x = 23 , y = 8 . It follows that h = 23 and k + 4 = 8 . Subtracting 4 from both sides of the equation k + 4 = 8 yields k = 4 . Substituting 23 for h and 4 for k in the equation y=a(x-h)2+k+4 yields y=a(x-23)2+4+4, or y=a(x-23)2+8. The value of a can be found by substituting any x-value and its corresponding y-value for x and y , respectively, in this equation. Substituting 25 for x and -8 for y in this equation yields -8=a(25-23)2+8, or -8=a(2)2+8. Subtracting 8 from both sides of this equation yields -16=a(2)2, or -16=4a. Dividing both sides of this equation by 4 yields -4=a. Substituting -4 for a , 23 for h , and 4 for k in the equation f(x)=a(x-h)2+k yields f(x)=-4(x-23)2+4. The y-intercept of the graph of y=f(x) in the xy-plane is the point on the graph where x = 0 . Substituting 0 for x in the equation f(x)=-4(x-23)2+4 yields f(0)=-4(0-23)2+4, or f(0)=-4(-23)2+4. This is equivalent to f(0)=-2,112, so the y-intercept of the graph of y=f(x) in the xy-plane is (0,-2,112). Thus, the y-coordinate of the y-intercept of the graph of y=f(x) in the xy-plane is -2,112.

Question 336 336 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

x225=36

What is a solution to the given equation?

  1. 6

  2. 30

  3. 450

  4. 900

Show Answer Correct Answer: B

Choice B is correct. Multiplying the left- and right-hand sides of the given equation by 25 yields x2=900. Taking the square root of the left- and right-hand sides of this equation yields x=30 or x=-30. Of these two solutions, only 30 is given as a choice.

Choice A is incorrect. This is a solution to the equation x2=36.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 337 337 of 479 selected Nonlinear Functions M

g(x)=11 ( 1 12 ) x

If the given function g is graphed in the xy-plane, where y=g(x), what is the y-intercept of the graph?

  1. (0,11)

  2. (0,132)

  3. (0,1)

  4. (0,12)

Show Answer Correct Answer: A

Choice A is correct. The x-coordinate of any y-intercept of a graph is 0 . Substituting 0 for x in the given equation yields g(0)=11(112)0. Since any nonzero number raised to the 0th power is 1 , this gives g(0)=11·1, or g(0)=11. The y-intercept of the graph is, therefore, the point (0,11).

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 338 338 of 479 selected Equivalent Expressions E

Which expression is equivalent to 9x2+7x2+9x?

  1. 63 x 4 + 9 x

  2. 9 x 2 + 16 x

  3. 25 x 5

  4. 16 x 2 + 9 x

Show Answer Correct Answer: D

Choice D is correct. In the given expression, the first two terms, 9x2 and 7x2, are like terms. Combining these like terms yields 9x2+7x2, or 16x2. It follows that the expression 9x2+7x2+9x is equivalent to 16x2+9x.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 339 339 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

P=N(19-C)

The given equation relates the positive numbers P , N , and C . Which equation correctly expresses C in terms of P and N ?

  1. C=19+PN

  2. C=19-PN

  3. C=19+PN

  4. C=19-PN

Show Answer Correct Answer: D

Choice D is correct. It's given that the values of P , N , and C are positive. Therefore, dividing each side of the given equation by N yields PN=19-C. Subtracting 19 from each side of this equation yields PN-19=-C. Dividing each side of this equation by -1 yields 19-PN=C, or C=19-PN.

Choice A is incorrect. This equation is equivalent to P=NC-19, not P=N(19-C).

Choice B is incorrect. This equation is equivalent to P=19-NC, not P=N(19-C).

Choice C is incorrect. This equation is equivalent to P=N(C-19), not P=N(19-C).

Question 340 340 of 479 selected Equivalent Expressions E

Which expression is equivalent to (2x2+x-9)+(x2+6x+1)?

  1. 2 x 2 + 7 x + 10

  2. 2 x 2 + 6 x - 8

  3. 3 x 2 + 7 x - 10

  4. 3 x 2 + 7 x - 8

Show Answer Correct Answer: D

Choice D is correct. The given expression is equivalent to (2x2+x+(-9))+(x2+6x+1), which can be rewritten as (2x2+x2)+(x+6x)+(-9+1). Adding like terms in this expression yields 3x2+7x+(-8), or 3x2+7x-8.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 341 341 of 479 selected Nonlinear Functions M

What is an x-coordinate of an x-intercept of the graph of y=3(x-14)(x+5)(x+4) in the xy-plane?

Show Answer Correct Answer: 14, -5, -4

The correct answer is either 14 , - 5 , or - 4 . The x-intercepts of a graph in the xy-plane are the points at which the graph intersects the x-axis, or when the value of y is 0 . Substituting 0 for y in the given equation yields 0=3(x-14)(x+5)(x+4). Dividing both sides of this equation by 3 yields 0=(x-14)(x+5)(x+4). Applying the zero product property to this equation yields three equations: x - 14 = 0 , x + 5 = 0 , and x + 4 = 0 . Adding 14 to both sides of the equation x - 14 = 0 yields x = 14 , subtracting 5 from both sides of the equation x + 5 = 0 yields x = - 5 , and subtracting 4 from both sides of the equation x + 4 = 0 yields x = - 4 . Therefore, the x-coordinates of the x-intercepts of the graph of the given equation are 14 , - 5 , and - 4 . Note that 14, -5, and -4 are examples of ways to enter a correct answer.

Question 342 342 of 479 selected Nonlinear Functions M

The area A, in square centimeters, of a rectangular painting can be represented by the expression w(w+29), where w is the width, in centimeters, of the painting.  Which expression represents the length, in centimeters, of the painting?

  1. w

  2. 29

  3. (w+29)

  4. w(w+29)

Show Answer Correct Answer: C

Choice C is correct. It's given that the expression w(w+29) represents the area, in square centimeters, of a rectangular painting, where w is the width, in centimeters, of the painting. The area of a rectangle can be calculated by multiplying its length by its width. It follows that the length, in centimeters, of the painting is represented by the expression (w+29).

Choice A is incorrect. This expression represents the width, in centimeters, of the painting, not its length, in centimeters.

Choice B is incorrect. This is the difference between the length, in centimeters, and the width, in centimeters, of the painting, not its length, in centimeters.

Choice D is incorrect. This expression represents the area, in square centimeters, of the painting, not its length, in centimeters.

Question 343 343 of 479 selected Nonlinear Functions H

A quadratic function models the height, in feet, of an object above the ground in terms of the time, in seconds, after the object is launched off an elevated surface. The model indicates the object has an initial height of 10 feet above the ground and reaches its maximum height of 1,034 feet above the ground 8 seconds after being launched. Based on the model, what is the height, in feet, of the object above the ground 10 seconds after being launched?

  1. 234

  2. 778

  3. 970

  4. 1,014

Show Answer Correct Answer: C

Choice C is correct. It's given that a quadratic function models the height, in feet, of an object above the ground in terms of the time, in seconds, after the object is launched off an elevated surface. This quadratic function can be defined by an equation of the form f(x)=a(x-h)2+k, where f(x) is the height of the object x seconds after it was launched, and a , h , and k are constants such that the function reaches its maximum value, k , when x = h . Since the model indicates the object reaches its maximum height of 1,034 feet above the ground 8 seconds after being launched, f(x) reaches its maximum value, 1,034 , when x = 8 . Therefore, k = 1,034 and h = 8 . Substituting 8 for h and 1,034 for k in the function f(x)=a(x-h)2+k yields f(x)=a(x-8)2+1,034. Since the model indicates the object has an initial height of 10 feet above the ground, the value of f(x) is 10 when x = 0 . Substituting 0 for x and 10 for f(x) in the equation f(x)=a(x-8)2+1,034 yields 10=a(0-8)2+1,034, or 10=64a+1,034. Subtracting 1,034 from both sides of this equation yields 64a=-1,024. Dividing both sides of this equation by 64 yields a = -16 . Therefore, the model can be represented by the equation f(x)=-16(x-8)2+1,034. Substituting 10 for x in this equation yields f(10)=-16(10-8)2+1,034, or f(10)=970. Therefore, based on the model, 10 seconds after being launched, the height of the object above the ground is 970 feet.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 344 344 of 479 selected Nonlinear Functions E

y=-14x2+ 2 x + 29

The given equation models a company’s scheduled deliveries over 8 months, where y is the estimated number of scheduled deliveries x months after the end of May 2012 , where 0x8. Which statement is the best interpretation of the y-intercept of the graph of this equation in the xy-plane?

  1. At the end of May 2012 , the estimated number of scheduled deliveries was 0 .

  2. At the end of May 2012 , the estimated number of scheduled deliveries was 29 .

  3. At the end of June 2012 , the estimated number of scheduled deliveries was 0 .

  4. At the end of June 2012 , the estimated number of scheduled deliveries was 29 .

Show Answer Correct Answer: B

Choice B is correct. The y-intercept of a graph in the xy-plane is the point where x=0. For the given equation, the y-intercept of the graph in the xy-plane can be found by substituting 0 for x in the equation, which yields y=-14(0)2+2(0)+29, or y=29. Therefore, the y-intercept of the graph is (0,29). It’s given that y is the estimated number of scheduled deliveries x months after the end of May 2012. Therefore, x=0 represents 0 months after the end of May 2012, or the end of May 2012. Thus, the best interpretation of the y-intercept of the graph of this equation in the xy-plane is that at the end of May 2012, the estimated number of scheduled deliveries was 29.

Choice A is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 345 345 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

  • For the line in the system:
    • The line slants sharply down from left to right.
    • The line passes through the following points:
      • (negative 2 comma 6)
      • approximately (0 comma 3.8)
  • For the curve in the system:
    • Moving from left to right:
      • The curve passes from quadrant 2 to quadrant 1.
      • In quadrant 2:
        • The curve trends down sharply to the point (negative 2 comma 6).
        • The curve then trends down gradually to the y axis.
      • In quadrant 1, the curve trends down gradually.
    • As x increases, the curve approaches the line y equals 5.
    • The curve passes through the following points:
      • (negative 3 comma 8)
      • (negative 2 comma 6)

The graph of a system of a linear equation and a nonlinear equation is shown. What is the solution (x,y) to this system?

  1. (6,0)

  2. (-2,6)

  3. (0,-2)

  4. (0,0)

Show Answer Correct Answer: B

Choice B is correct. The solution (x,y) to the system of two equations corresponds to the point where the graphs of the equations intersect in the xy-plane. The graphs of the linear equation and the nonlinear equation shown intersect at the point (-2,6). Thus, the solution (x,y) to this system is (-2,6).

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 346 346 of 479 selected Equivalent Expressions H

The equation the fraction with numerator x squared, plus 6 x, minus 7, and denominator x plus 7, end fraction, equals, a x plus d is true for all x not equal to negative 7, where a and d are integers. What is the value of a plus d ?

  1. negative 6

  2. negative 1

  3. 0

  4. 1

Show Answer Correct Answer: C

Choice C is correct. Since the expression x squared, plus 6 x, minus 7 can be factored as open parenthesis, x plus 7, close parenthesis, times, open parenthesis, x minus 1, close parenthesis, the given equation can be rewritten as the fraction with numerator open parenthesis, x plus 7, close parenthesis, times, open parenthesis, x minus 1, close parenthesis, and denominator x plus 7, end fraction, equals a, x plus d. Since x is not equal to negative 7, x plus 7 is also not equal to 0, so both the numerator and denominator of the fraction with numerator open parenthesis, x plus 7, close parenthesis, times, open parenthesis, x minus 1, close parenthesis, and denominator x plus 7, end fraction can be divided by x plus 7. This gives x minus 1, equals a x plus d. Equating the coefficient of x on each side of the equation gives a, equals 1. Equating the constant terms gives d equals negative 1. The sum is 1 plus negative 1, equals 0.

Choice A is incorrect and may result from incorrectly simplifying the equation. Choices B and D are incorrect. They are the values of d and a, respectively, not a, plus d.

 

Question 347 347 of 479 selected Equivalent Expressions M

Which expression is equivalent to (8x3+8)-(x3-2)?

  1. 8 x 3 + 6

  2. 7 x 3 + 10

  3. 8 x 3 + 10

  4. 7 x 3 + 6

Show Answer Correct Answer: B

Choice B is correct. The given expression is equivalent to 8x3+8-x3-(-2), or  8x3+8-x3+2. Combining like terms in this expression yields 7x3+10.

Choice A is incorrect. This expression is equivalent to (8x3+8)-2, not (8x3+8)-(x3-2).

Choice C is incorrect. This expression is equivalent to (8x3+8)-(-2), not (8x3+8)-(x3-2).

Choice D is incorrect. This expression is equivalent to (8x3+8)-(x3+2), not (8x3+8)-(x3-2).

Question 348 348 of 479 selected Equivalent Expressions H

Which of the following expressions is equivalent to the fraction with numerator x squared, minus 2 x, minus 5, and denominator x minus 3 ?

  1. x minus 5, minus the fraction with numerator 20, and denominator x minus 3

  2. x minus 5, minus the fraction with numerator 10, and denominator x minus 3

  3. x plus 1, minus the fraction with numerator 8, and denominator x minus 3

  4. x plus 1, minus the fraction with numerator 2, and denominator x minus 3

Show Answer Correct Answer: D

Choice D is correct. The numerator of the given expression can be rewritten in terms of the denominator, x minus 3, as follows: x squared, minus 2 x, minus 5, equals, x squared, minus 3 x, plus x, minus 3, minus 2, which is equivalent to x times, open parenthesis, x minus 3, close parenthesis, plus, open parenthesis, x minus 3, close parenthesis, minus 2. So the given expression is equivalent to the fraction with numerator x times, open parenthesis, x minus 3, close parenthesis, plus, open parenthesis, x minus 3, close parenthesis, minus 2, and denominator x minus 3, end fraction, equals, the fraction with numerator x times, open parenthesis, x minus 3, close parenthesis, and denominator x minus 3, end fraction, plus the fraction with numerator x minus 3, and denominator x minus 3, end fraction, minus the fraction with numerator 2, and denominator x minus 3, end fraction. Since the given expression is defined for x is not equal to 3, the expression can be rewritten as x plus 1, minus, the fraction with numerator 2, and denominator x minus 3, end fraction.

Long division can also be used as an alternate approach. Choices A, B, and C are incorrect and may result from errors made when dividing the two polynomials or making use of structure.

 

Question 349 349 of 479 selected Nonlinear Functions M

A company opens an account with an initial balance of $36,100.00 . The account earns interest, and no additional deposits or withdrawals are made. The account balance is given by an exponential function A , where A(t) is the account balance, in dollars, t years after the account is opened. The account balance after 13 years is $68,071.93 . Which equation could define A ?

  1. A(t)=36,100.00 (1.05)t

  2. A(t)=31,971.93 (1.05)t

  3. A(t)=31,971.93 (0.05)t

  4. A(t)=36,100.00 (0.05)t

Show Answer Correct Answer: A

Choice A is correct. Since it's given that the account balance, A(t), in dollars, after t years can be modeled by an exponential function, it follows that function A can be written in the form A(t)=Nrt, where N is the initial value of the function and r is a constant related to the growth of the function. It's given that the initial balance of the account is $36,100.00, so it follows that the initial value of the function, or N , must be 36,100.00. Substituting 36,100.00 for N in the equation A(t)=Nrt yields A(t)=36,100.00rt. It's given that the account balance after 13 years, or when t=13, is $68,071.93. It follows that A(13)=68,071.93, or 36,100.00r13=68,071.93. Dividing each side of the equation 36,100.00r13=68,071.93 by 36,100.00 yields r13=68,071.9336,100.00. Taking the 13 th root of both sides of this equation yields r=68,071.9336,100.0013, or r is approximately equal to 1.05 . Substituting 1.05 for r in the equation A(t)=36,100.00rt yields A(t)=36,100.00(1.05)t, so the equation A(t)=36,100.00(1.05)t could define A

Choice B is incorrect. Substituting 0 for t in this function indicates an initial balance of $31,971.93, rather than $36,100.00.

Choice C is incorrect. Substituting 0 for t in this function indicates an initial balance of $31,971.93, rather than $36,100.00. Additionally, this function indicates the account balance is decreasing, rather than increasing, over time. 

Choice D is incorrect. This function indicates the account balance is decreasing, rather than increasing, over time.

Question 350 350 of 479 selected Equivalent Expressions M

Which of the following is equivalent to the sum of 3 x to the fourth power, plus 2 x cubed and 4 x to the fourth power, plus 7 x cubed?

  1. 16 x to the fourteenth power

  2. 7 x to the eighth power, plus 9 x to the sixth power

  3. 12 x to the fourth power, plus 14 x cubed

  4. 7 x to the fourth power, plus 9 x cubed

Show Answer Correct Answer: D

Choice D is correct. Adding the two expressions yields 3 x to the fourth power, plus 2 x to the third power, plus 4 x to the fourth power, plus 7 x to the third power. Because the pair of terms 3 x to the fourth power and 4 x to the fourth power and the pair of terms 2 x to the third power and 7 x to the third power each contain the same variable raised to the same power, they are like terms and can be combined as 7 x to the fourth power and 9 x to the third power, respectively. The sum of the given expressions therefore simplifies to 7 x to the fourth power, plus 9 x to the third power.

Choice A is incorrect and may result from adding the coefficients and the exponents in the given expressions. Choice B is incorrect and may result from adding the exponents as well as the coefficients of the like terms in the given expressions. Choice C is incorrect and may result from multiplying, rather than adding, the coefficients of the like terms in the given expressions.

 

Question 351 351 of 479 selected Nonlinear Functions E

An investment account was opened with an initial value of $890. The value of the account doubled every 10 years. Which equation represents the value of the account M(t), in dollars, t years after the account was opened?

  1. M(t)=890(12)t10

  2. M(t)=890(110)t2

  3. M(t)=890(2)t10

  4. M(t)=890(10)t2

Show Answer Correct Answer: C

Choice C is correct. It's given that t represents the number of years since the account was opened. Therefore, t10 represents the number of 10 -year periods since the account was opened. Since the value of the account doubles during each of these 10 -year periods, the value of the account can be found by multiplying the initial value by t10 factors of 2 . This is equivalent to 2t10. It's given that the initial value of the account is $890. Therefore, the value of the account M(t), in dollars, t years after the account was opened can be represented by M(t)=890(2)t10.

Choice A is incorrect. This equation represents the value of an account if the value of the account halves, not doubles, every 10 years.

Choice B is incorrect. This equation represents the value of an account if the value of the account decreases by 90%, not doubles, every 2 , not 10 , years.

Choice D is incorrect. This equation represents the value of an account if the value of the account increases by a factor of 10 , not doubles, every 2 , not 10 , years.

Question 352 352 of 479 selected Equivalent Expressions H

open parenthesis, 7,532, plus 100 y squared, close parenthesis, plus, 10 times open parenthesis, 10 y squared minus 110, close parenthesis

The expression above can be written in the form a, times y squared, plus b, where a and b are constants. What is the value of a, plus b ?

Show Answer

The correct answer is 6632. Applying the distributive property to the expression yields open parenthesis, 7532 plus 100 y squared, close parenthesis, plus, open parenthesis, 100 y squared, minus 1100, close parenthesis. Then adding together 7532 plus 100 y squared and 100 y squared, minus 1100 and collecting like terms results in 200 y squared, plus 6432. This is written in the form a, y squared, plus b, where a, equals 200 and b equals 6432. Therefore a, plus b equals, 200 plus 6432, which equals 6632.

Question 353 353 of 479 selected Nonlinear Functions E

The y -intercept of the graph of y = x 2 + 31 in the xy-plane is (0,y). What is the value of y ?

Show Answer Correct Answer: 31

The correct answer is 31 . It's given that the y-intercept of the graph of y = x 2 + 31 in the xy-plane is (0,y). Substituting 0 for x in the given equation yields y=(0)2+31, or y = 31 . Thus, the value of y is 31 .

Question 354 354 of 479 selected Nonlinear Functions M
xf(x)
05
1five halves
2five fourths
3five eighths

The table above gives the values of the function f for some values of x. Which of the following equations could define f ?

  1. f of x equals, 5 times, open parenthesis, 2 raised to the x plus 1 power, close parenthesis

  2. f of x equals, 5 times, open parenthesis, 2 to the x power, close parenthesis

  3. f of x equals, 5 times, open parenthesis, 2 raised to the negative, open parenthesis, x plus 1, close parenthesis, power, close parenthesis

  4. f of x equals, 5 times, open parenthesis, 2 to the negative x power, close parenthesis

Show Answer Correct Answer: D

Choice D is correct. Each choice has a function with coefficient 5 and base 2, so the exponents must be analyzed. When the input value of x increases, the output value of f(x) decreases, so the exponent must be negative. An exponent of –x yields the values in the table: first value, 5, which equals 5 times open parenthesis, 2 to the negative 0 power, close parenthesis, second value, five halves, which equals 5 times open parenthesis, 2 to the negative 1 power, close parenthesis, third value, five fourths, which equals 2 to the negative 2 power, and fourth value, five eighths, which equals 5 times open parenthesis, 2 to the negative 3 power, close parenthesis.

Choices A and B are incorrect and may result from choosing equations that yield an increasing, rather than decreasing, output value of f(x) when the input value of x increases. Choice C is incorrect and may result from choosing an equation that doesn’t yield the values in the table.

 

Question 355 355 of 479 selected Equivalent Expressions H

The expression 635x455·28x8 is equivalent to a x b , where a and b are positive constants and x>1. What is the value of a + b ?

Show Answer Correct Answer: 361/8, 45.12, 45.13

The correct answer is 3618. The rational exponent property is ymn=ymn, where y>0, m and n are integers, and n>0. This property can be applied to rewrite the given expression 635x455·28x8 as 6(355)(x455)(288)(x18), or 6(3)(x9)(2)(x18). This expression can be rewritten by multiplying the constants, which gives 36(x9)(x18). The multiplication exponent property is yn·ym=yn+m, where y>0. This property can be applied to rewrite the expression 36(x9)(x18) as 36x9+18, or 36x738. Therefore, 635x455·28x8=36x738. It's given that 635x455·28x8 is equivalent to axb; therefore, a=36 and b=738. It follows that a+b=36+738. Finding a common denominator on the right-hand side of this equation gives a+b=2888+738, or a+b=3618. Note that 361/8, 45.12, and 45.13 are examples of ways to enter a correct answer.

Question 356 356 of 479 selected Equivalent Expressions M

2, x squared, plus 5 x, minus 12

If the given expression is rewritten in the form open parenthesis, 2 x minus 3, close parenthesis, times, open parenthesis, x plus k, close parenthesis, where k is a constant, what is the value of k ?

Show Answer

The correct answer is 4. It’s given that 2 x squared, plus 5 x, minus 12 can be rewritten as open parenthesis, 2 x minus 3, close parenthesis, times, open parenthesis, x plus k, close parenthesis; it follows that open parenthesis, 2 x minus 3, close parenthesis, times, open parenthesis, x plus k, close parenthesis, equals, 2 x squared, plus 5 x, minus 12. Expanding the left-hand side of this equation yields 2 x squared, plus 2 k x, minus 3 x, minus 3 k, equals, 2 x squared, plus 5 x, minus 12. Subtracting 2 x squared from both sides of this equation yields 2 k x minus 3 x, minus 3 k, equals, 5 x minus 12. Using properties of equality, 2 k x minus 3 x, equals 5 x and negative 3 k, equals negative 12. Either equation can be solved for k. Dividing both sides of negative 3 k, equals negative 12 by negative 3 yields k equals 4. The equation 2 k x minus 3 x, equals 5 x can be rewritten as x times, open parenthesis, 2 k minus 3, close parenthesis, equals 5 x. It follows that 2 k minus 3, equals 5. Solving this equation for k also yields k equals 4. Therefore, the value of k is 4.

Question 357 357 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

v squared equals, the fraction with numerator L T, and denominator m

The formula above expresses the square of the speed v of a wave moving along a string in terms of tension T, mass m, and length L of the string. What is T in terms of m, v, and L ?

  1. T equals, the fraction with numerator m v squared, and denominator L
  2. T equals, the fraction with numerator m, and denominator v squared L, end fraction
  3. T equals, the fraction with numerator m L, and denominator v squared, end fraction
  4. T equals, the fraction with numerator L, and denominator m v squared, end fraction
Show Answer Correct Answer: A

Choice A is correct. To write the formula as T in terms of m, v, and L means to isolate T on one side of the equation. First, multiply both sides of the equation by m, which gives m v squared, equals the fraction with numerator m L T, and denominator m, which simplifies to mv2 = LT. Next, divide both sides of the equation by L, which gives the fraction with numerator m v squared, and denominator L, equals the fraction with numerator L T, and denominator L, which simplifies to T equals the fraction with numerator m v squared, and denominator L.

Choices B, C, and D are incorrect and may be the result of incorrectly applying operations to each side of the equation.

Question 358 358 of 479 selected Nonlinear Functions M

The function is defined by f(x)=5(14-x)2+114. What is the value of f(14)?

Show Answer Correct Answer: 11/4, 2.75

The correct answer is 114. It’s given that the function f is defined by f(x)=5(14-x)2+114. Substituting 14 for x in this equation yields f(14)=5(14-14)2+114, which is equivalent f(14)=5(0)2+114, or f(14)=114. Therefore, the value of f(14) is 114. Note that 11/4 or 2.75 are examples of ways to enter a correct answer.

Question 359 359 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

64x2-(16a+4b)x+ab=0

In the given equation, a and b are positive constants. The sum of the solutions to the given equation is  k(4a+b), where k is a constant. What is the value of k ?

Show Answer Correct Answer: .0625, 1/16

The correct answer is 1 16 . Let p and q represent the solutions to the given equation. Then, the given equation can be rewritten as 64(x-p)(x-q)=0, or 64x2-64(p+q)+pq=0. Since this equation is equivalent to the given equation, it follows that -(16a+4b)=-64(p+q). Dividing both sides of this equation by -64 yields 16a+4b64=p+q, or 116(4a+b)=p+q. Therefore, the sum of the solutions to the given equation, p+q, is equal to 116(4a+b). Since it's given that the sum of the solutions to the given equation is k(4a+b), where k is a constant, it follows that k = 1 16 . Note that 1/16, .0625, 0.062, and 0.063 are examples of ways to enter a correct answer.

Alternate approach: The given equation can be rewritten as 64x2-4(4a+b)x+ab=0, where a and b are positive constants. Dividing both sides of this equation by 4 yields 16x2-(4a+b)x+ab4=0. The solutions for a quadratic equation in the form Ax2+Bx+C=0, where A , B , and C are constants, can be calculated using the quadratic formula, x=-B+B2-4AC2A and x=-B-B2-4AC2A. It follows that the sum of the solutions to a quadratic equation in the form A x 2 + B x + C = 0 is -B+B2-4AC2A+-B-B2-4AC2A, which can be rewritten as -B+-B+B2-4AC-B2-4AC2A, which is equivalent to -2B2A, or - B A . In the equation 16x2-(4a+b)x+ab4=0, A = 16 B=-(4a+b), and C = a b 4 . Substituting 16 for A and -(4a+b) for B in - B A yields --(4a+b)16, which can be rewritten as 116(4a+b). Thus, the sum of the solutions to the given equation is 116(4a+b). Since it's given that the sum of the solutions to the given equation is k(4a+b), where k is a constant, it follows that k = 1 16

Question 360 360 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

| x - 2 | = 9

What is one possible solution to the given equation?

Show Answer Correct Answer: 11, -7

The correct answer is 11 or -7 . By the definition of absolute value, if |x-2|=9, then x-2=9 or x-2=-9. Adding 2 to both sides of the equation x-2=9 yields x=11. Adding 2 to both sides of the equation x-2=-9 yields x=-7. Thus, the given equation, |x-2|=9, has two possible solutions, 11 and -7 . Note that 11 and -7 are examples of ways to enter a correct answer.

Question 361 361 of 479 selected Nonlinear Functions E

The graph shows the predicted value y , in dollars, of a certain sport utility vehicle x years after it is first purchased.

  • The curve is in quadrant 1.
  • The curve trends down gradually from left to right.
  • The curve passes through the following approximate points:
    • (0 comma 23,000)
    • (1 comma 19,320)
    • (3 comma 13,632)
    • (5 comma 9,619)

Which of the following is closest to the predicted value of the sport utility vehicle 3 years after it is first purchased?

  1. $9,619

  2. $13,632

  3. $19,320

  4. $23,000

Show Answer Correct Answer: B

Choice B is correct. For the graph shown, the horizontal axis represents the number of years after a certain sport utility vehicle is first purchased, and the vertical axis represents the predicted value, in dollars, of the sport utility vehicle. According to the graph, 3 years after the sport utility vehicle is purchased, the predicted value of the sport utility vehicle is between $10,000 and $15,000. Of the given choices, only $13,632 is between $10,000 and $15,000. Therefore, $13,632 is closest to the predicted value of the sport utility vehicle 3 years after it is first purchased.

Choice A is incorrect. This is closest to the predicted value of the sport utility vehicle 5 years after it is first purchased.

Choice C is incorrect. This is closest to the predicted value of the sport utility vehicle 1 year after it is first purchased.

Choice D is incorrect. This is closest to the predicted value of the sport utility vehicle when it is first purchased.

Question 362 362 of 479 selected Equivalent Expressions M

Which expression represents the product of (x-6y3z5) and (x4z5+y8z-7)?

  1. x-2z10+y11z-2

  2. x-2z10+x-6z-2

  3. x-2y3z10+y8z-7

  4. x-2y3z10+x-6y11z-2

Show Answer Correct Answer: D

Choice D is correct. The product of (x-6y3z5) and (x4z5+y8z-7) can be represented by the expression (x-6y3z5)(x4z5+y8z-7). Applying the distributive property to this expression yields (x-6y3z5)(x4z5)+(x-6y3z5)(y8z-7), or x-6x4y3z5z5+x-6y3y8z5z-7. This expression is equivalent to x-6+4y3z5+5+x-6y3+8z5-7, or x-2y3z10+x-6y11z-2.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 363 363 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M
y equals x squared
2 y plus 6, equals, 2 times, open parenthesis, x plus 3, close
parenthesis

If (x, y) is a solution of the system of equations above and x > 0, what is the value of xy ?

  1. 1

  2. 2

  3. 3

  4. 9

Show Answer Correct Answer: A

Choice A is correct. Substituting x squared for y in the second equation gives 2 times, x squared, plus 6, equals, 2 times, open parenthesis, x plus 3, close parenthesis. This equation can be solved as follows:

2 x squared, plus 6, equals, 2 x plus 6Apply the distributive property.
2 x squared, plus 6, minus 2 x, minus 6, equals 0Subtract 2x and 6 from both sides of the equation.
2 x squared, minus 2 x, equals 0Combine like terms.
2 x times, open parenthesis, x minus 1, close parenthesis, equals 0Factor both terms on the left side of the equation
by 2x.

Thus, x equals 0 and x equals 1 are the solutions to the system. Since x is greater than 0, only x equals 1 needs to be considered. The value of y when x equals 1 is y equals x squared, which equals 1 squared, which equals 1. Therefore, the value of xy is 1 times 1, equals 1.

Choices B, C, and D are incorrect and likely result from a computational or conceptual error when solving this system of equations.

 

Question 364 364 of 479 selected Nonlinear Functions H

Function f is defined by f(x)=(x+6)(x+5)(x+1). Function g is defined by g(x)=f(x-1). The graph of y=g(x) in the xy-plane has x-intercepts at (a,0)(b,0), and (c,0), where a , b , and c are distinct constants. What is the value of a + b + c ?

  1. -15

  2. -9

  3. 11

  4. 15

Show Answer Correct Answer: B

Choice B is correct. It's given that g(x)=f(x-1). Since f(x)=(x+6)(x+5)(x+1), it follows that f(x-1)=(x-1+6)(x-1+5)(x-1+1). Combining like terms yields f(x-1)=(x+5)(x+4)(x). Therefore, g(x)=x(x+5)(x+4). The x-intercepts of a graph in the xy-plane are the points where y = 0 . The x-coordinates of the x-intercepts of the graph of y=g(x) in the xy-plane can be found by solving the equation 0=x(x+5)(x+4). Applying the zero product property to this equation yields three equations:  x = 0 , x + 5 = 0 , and x + 4 = 0 . Solving each of these equations for x yields x = 0 , x = -5 , and x = -4 , respectively. Therefore, the x-intercepts of the graph of y=g(x) are (0,0)(-5,0), and (-4,0). It follows that the values of a , b , and c are 0 , -5 , and -4 . Thus, the value of a + b + c is 0+(-5)+(-4), which is equal to -9 .

Choice A is incorrect. This is the value of a + b + c if g(x)=f(x+1).

Choice C is incorrect. This is the value of a + b + c - 1 if g(x)=(x-6)(x-5)(x-1).

Choice D is incorrect. This is the value of a + b + c if f(x)=(x-6)(x-5)(x-1).

Question 365 365 of 479 selected Equivalent Expressions H

If a equals, c plus d, which of the following is equivalent to the expression x squared, minus c squared, minus 2 c d, minus d squared?

  1. open parenthesis, x plus a, close parenthesis, squared
  2. open parenthesis, x minus a, close parenthesis, squared
  3. open parenthesis, x plus a, close parenthesis, times, open parenthesis, x minus a, close parenthesis
  4. x squared, minus a times x, minus a squared
Show Answer Correct Answer: C

Choice C is correct. Factoring 1 from the second, third, and fourth terms gives x2 c2 – 2cd d2 = x2 – (c2 + 2cd + d2). The expression c2 + 2cd + d2 is the expanded form of a perfect square: c2 + 2cd + d2 = (c + d)2. Therefore, x2 – (c2 + 2cd + d2) = x2 – (c + d)2. Since a = c + d, x2 – (c + d)2 = x2 a2. Finally, because x2 a2 is the difference of squares, it can be expanded as x2 a2 = (x + a)(x a).

Choices A and B are incorrect and may be the result of making an error in factoring the difference of squares x2 a2. Choice D is incorrect and may be the result of incorrectly combining terms.

Question 366 366 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

x 2 - 40 x - 10 = 0

What is the sum of the solutions to the given equation?

  1. 0

  2. 5

  3. 10

  4. 40

Show Answer Correct Answer: D

Choice D is correct. Adding 10 to each side of the given equation yields x 2 - 40 x = 10 . To complete the square, adding (402)2, or 202, to each side of this equation yields x2-40x+202=10+202, or ( x - 20 ) 2 = 410 . Taking the square root of each side of this equation yields x-20=±410. Adding 20 to each side of this equation yields x=20±410. Therefore, the solutions to the given equation are x=20+410 and x=20-410. The sum of these solutions is (20+410)+(20-410), or 40 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 367 367 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

x = 8

y = x 2 + 8

The graphs of the equations in the given system of equations intersect at the point (x,y) in the xy-plane. What is the value of y ?

  1. 8

  2. 24

  3. 64

  4. 72

Show Answer Correct Answer: D

Choice D is correct. Since the graphs of the equations in the given system intersect at the point (x,y), the point (x,y) represents a solution to the given system of equations. The first equation of the given system of equations states that x = 8 . Substituting 8 for x in the second equation of the given system of equations yields y=82+8, or y = 72 . Therefore, the value of y is 72 .

Choice A is incorrect. This is the value of x , not y .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 368 368 of 479 selected Nonlinear Functions E

The function g is defined by g(x)=|x+18|. What is the value of g(4)?

  1. -18

  2. -4

  3. 14

  4. 22

Show Answer Correct Answer: D

Choice D is correct. The value of g(4) is the value of g(x) when x = 4 . Substituting 4 for x in the given equation yields g(4)=|4+18|, which is equivalent to g(4)=|22|, or g(4)=22. Therefore, the value of g(4) is 22 .

Choice A is incorrect. This would be the value of g(4) if function g was defined by g(x)=-|18|, not g(x)=|x+18|.

Choice B is incorrect. This would be the value of g(4) if function g was defined by g(x)=-|x|, not g(x)=|x+18|.

Choice C is incorrect. This would be the value of g(4) if function g was defined by g(x)=|-x+18|, not g(x)=|x+18|.

Question 369 369 of 479 selected Equivalent Expressions E

open parenthesis, 2 x cubed, plus 3 x, close parenthesis, times, open parenthesis, x cubed, minus 2 x, close parenthesis

Which of the following is equivalent to the expression above?

  1. x cubed, plus 5 x

  2. 3 x cubed, plus x

  3. 2 x to the sixth power, minus x to the fourth power, minus 6 x squared

  4. 3 x to the sixth power, minus x to the fourth power, minus 6 x squared

Show Answer Correct Answer: C

Choice C is correct. Using the distributive property to multiply the terms in the parentheses yields open parenthesis, 2 x cubed, times x cubed, close parenthesis, plus, open parenthesis, 2 x cubed, times negative 2 x, close parenthesis, plus, open parenthesis, 3 x times x cubed, close parenthesis, plus, open parenthesis, 3 x times negative 2 x, close parenthesis, which is equivalent to 2 x to the sixth power minus 4 x to the fourth power, plus 3 x to the fourth power, minus 6 x squared. Combining like terms results in  2 x to the sixth power minus x to the fourth power, minus 6 x squared.

Choices A and D are incorrect and may result from conceptual errors when multiplying the terms in the given expression. Choice B is incorrect and may result from adding, instead of multiplying, open parenthesis, 2 x cubed, plus 3 x, close parenthesis and open parenthesis, x cubed, minus 2 x, close parenthesis.

Question 370 370 of 479 selected Nonlinear Functions M

The exponential function g is defined by g(x)=19·ax, where a is a positive constant. If g(3)=2,375 , what is the value of g(4)?

Show Answer Correct Answer: 11875

The correct answer is 11,875 . It's given that the exponential function g is defined by g(x)=19·ax, where a is a positive constant, and g(3)=2,375. It follows that when x = 3 , g(x)=2,375. Substituting 3 for x and 2,375 for g(x) in the given equation yields 2,375=19·a3. Dividing each side of this equation by 19 yields 125=a3. Taking the cube root of both sides of this equation gives a = 5 . Substituting 4 for x and 5 for a in the equation g(x)=19·ax yields g(4)=19·54, or g(4)=11,875. Therefore, the value of g(4) is 11,875

Question 371 371 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

55 x + 6 = x

What is the positive solution to the given equation?

Show Answer Correct Answer: 5

The correct answer is 5 . Multiplying both sides of the given equation by x + 6 results in 55=x(x+6). Applying the distributive property of multiplication to the right-hand side of this equation results in 55 = x 2 + 6 x . Subtracting 55 from both sides of this equation results in 0 = x 2 + 6 x - 55 . The right-hand side of this equation can be rewritten by factoring. The two values that multiply to -55 and add to 6 are 11 and -5 . It follows that the equation 0 = x 2 + 6 x - 55 can be rewritten as 0=(x+11)(x-5). Setting each factor equal to 0 yields two equations: x + 11 = 0 and x - 5 = 0 . Subtracting 11 from both sides of the equation x + 11 = 0 results in x = -11 . Adding 5 to both sides of the equation x - 5 = 0 results in x = 5 . Therefore, the positive solution to the given equation is 5 .

Question 372 372 of 479 selected Nonlinear Functions M

An egg is thrown from a rooftop. The equation h=-4.9t2+9t+18 represents this situation, where h is the height of the egg above the ground, in meters, t seconds after it is thrown. According to the equation, what is the height, in meters, from which the egg was thrown?

Show Answer Correct Answer: 18

The correct answer is 18 . It's given that an egg is thrown from a rooftop and that the equation h=-4.9t2+9t+18 represents this situation, where h is the height of the egg above the ground, in meters, t seconds after it is thrown. If follows that the height, in meters, from which the egg was thrown is the value of h when t = 0 . Substituting 0 for t in the equation h=-4.9t2+9t+18 yields h(0)=-4.9(0)2+9(0)+18, or h = 18 . Therefore, according to the equation, the height, in meters, from which the egg was thrown is 18 .

Question 373 373 of 479 selected Nonlinear Functions H

The total distance d, in meters, traveled by an object moving in a straight line can be modeled by a quadratic function that is defined in terms of t, where t is the time in seconds. At a time of 10.0 seconds, the total distance traveled by the object is 50.0 meters, and at a time of 20.0 seconds, the total distance traveled by the object is 200.0 meters. If the object was at a distance of 0 meters when t equals 0, then what is the total distance traveled, in meters, by the object after 30.0 seconds?

Show Answer

The correct answer is 450. The quadratic equation that models this situation can be written in the form d equals, a, t squared, plus b t, plus c, where a, b, and c are constants. It’s given that the distance, d, the object traveled was 0 meters when t equals 0 seconds. These values can be substituted into the equation to solve for a, b, and c: 0 equals, a, times 0 squared, plus, b times 0, plus c. Therefore, c equals 0, and it follows that d equals, a, t squared, plus b t. Since it’s also given that d is 50 when t is 10 and d is 200 when t is 20, these values for d and t can be substituted to create a system of two linear equations: 50 equals, a, times 10 squared, plus, b times 10 and 200 equals, a, times 20 squared, plus, b times 20, or 10 a, plus b, equals 5 and 20 a, plus b, equals 10. Subtracting the first equation from the second equation yields 10 a, equals 5, or a, equals one half. Substituting one half for a in the first equation and solving for b yields b equals 0. Therefore, the equation that represents this situation is d equals, one half t squared. Evaluating this function when t equals 30 seconds yields d, equals, one half open parenthesis, 30, close parenthesis, squared, equals 450, or d equals 450 meters.

Question 374 374 of 479 selected Equivalent Expressions M

Which of the following is equivalent to  ?

  1. 1 minus p to the power 8

  2. 1 minus p to the power 7

  3. 1 minus p to the power 6

  4. 1 minus p to the power 5

Show Answer Correct Answer: B

Choice B is correct. Multiplying (1 – p) by each term of the polynomial within the second pair of parentheses gives (1 – p)1 = 1 – p; (1 – p)p = pp2; (1 – p)p2 = p2p3; (1 – p)p3 = p3p4; (1 – p)p4 = p4p5; (1 – p)p5 = p5p6; and (1 – p)p6 = p6p7. Adding these seven expressions together and combining like terms gives 1 + (p – p) + (p2 – p2) + (p3 – p3) + (p4 – p4) + (p5 – p5) + (p6 – p6) – p7, which can be simplified to 1 – p7.


Choices A, C, and D are incorrect and may result from incorrectly identifying the highest power of p in the expressions or incorrectly combining like terms.

 

Question 375 375 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

y equals, x squared, plus 3 x, minus 7, and, y minus 5 x, plus 8, equals 0

How many solutions are there to the system of equations above?

  1. There are exactly 4 solutions.

  2. There are exactly 2 solutions.

  3. There is exactly 1 solution.

  4. There are no solutions.

Show Answer Correct Answer: C

Choice C is correct. The second equation of the system can be rewritten as y equals, 5 x minus 8. Substituting 5 x minus 8 for y in the first equation gives 5 x minus 8, equals, x squared, plus 3 x, minus 7. This equation can be solved as shown below:

x squared, plus 3 x, minus 7, minus 5 x, plus 8, equals 0

x squared, minus 2 x, plus 1, equals 0

open parenthesis, x minus 1, close parenthesis, squared, equals 0

x equals 1

Substituting 1 for x in the equation y equals, 5 x minus 8 gives y equals negative 3. Therefore, the ordered pair 1 comma negative 3 is the only solution to the system of equations.

Choice A is incorrect. In the xy-plane, a parabola and a line can intersect at no more than two points. Since the graph of the first equation is a parabola and the graph of the second equation is a line, the system cannot have more than 2 solutions. Choice B is incorrect. There is a single ordered pair x comma y that satisfies both equations of the system. Choice D is incorrect because the ordered pair 1 comma negative 3 satisfies both equations of the system.

 

Question 376 376 of 479 selected Nonlinear Functions H

A submersible device is used for ocean research. The function g(x)=-155(x+19)(x-35) gives the depth below the surface of the ocean, in meters, of the submersible device x minutes after collecting a sample, where x>0. How many minutes after collecting the sample did it take for the submersible device to reach the surface of the ocean?

Show Answer Correct Answer: 35

The correct answer is 35 . It's given that the function g(x)=-155(x+19)(x-35) gives the depth below the surface of the ocean, in meters, of the submersible device x minutes after collecting a sample, where x>0. It follows that when the submersible device is at the surface of the ocean, the value of g(x) is 0 . Substituting 0 for g(x) in the equation g(x)=-155(x+19)(x-35) yields 0=-155(x+19)(x-35). Multiplying both sides of this equation by - 55 yields 0=(x+19)(x-35). Since a product of two factors is equal to 0 if and only if at least one of the factors is 0 , either x + 19 = 0 or x - 35 = 0 . Subtracting 19 from both sides of the equation x + 19 = 0 yields x = - 19 . Adding 35 to both sides of the equation x - 35 = 0 yields x = 35 . Since x>0, 35 minutes after collecting the sample the submersible device reached the surface of the ocean.

Question 377 377 of 479 selected Equivalent Expressions M

open parenthesis, 2 x plus 3, close parenthesis, minus, open parenthesis, x minus 7, close parenthesis

Which of the following is equivalent to the given expression?

  1. x minus 4

  2. 3 x minus 4

  3. x plus 10

  4. 2 x squared, plus 21

Show Answer Correct Answer: C

Choice C is correct. Distributing the negative sign to the terms in the second parentheses yields open parenthesis, 2 x plus 3, close parenthesis, minus x, plus 7. This expression can be rewritten as 2 x minus x, plus 3, plus 7. Combining like terms results in x plus 10.

Choice A is incorrect and may result from not distributing the negative sign to the 7. Choice B is incorrect and may result from adding x minus 7 to 2 x plus 3 instead of subtracting x minus 7. Choice D is incorrect and may result from adding the product of 2 x and x to the product of 3 and 7.

 

Question 378 378 of 479 selected Nonlinear Functions M

The function p is defined by p(n)=7n3. What is the value of n when p(n) is equal to 56 ?

  1. 2

  2. 8 3

  3. 7

  4. 8

Show Answer Correct Answer: A

Choice A is correct. It's given that p(n)=7n3. Substituting 56 for p(n) in this equation yields 56=7n3. Dividing each side of this equation by 7 yields 8=n3. Taking the cube root of each side of this equation yields 2=n. Therefore, when p(n) is equal to 56 , the value of n is 2 .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 379 379 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

6 x squared, plus 5 x, minus 7, equals 0

What are the solutions to the given equation?

  1. the fraction with numerator negative 5, plus or minus the square root of 25 plus 168, end root, and denominator 12

  2. the fraction with numerator negative 6, plus or minus the square root of 25 plus 168, end root, and denominator 12

  3. the fraction with numerator negative 5, plus or minus the square root of 36 minus 168, end root, and denominator 12

  4. the fraction with numerator negative 6, plus or minus the square root of 36 minus 168, end root, and denominator 12

Show Answer Correct Answer: A

Choice A is correct. The quadratic formula, x equals, the fraction with numerator negative b plus or minus, the square root of, b squared, minus 4 a, c, end root, and denominator 2 a, end fraction, can be used to find the solutions to an equation in the form a, x squared, plus b x, plus c, equals 0. In the given equation, a, equals 6, b equals 5, and c equals negative 7. Substituting these values into the quadratic formula gives the fraction with numerator negative 5 plus or minus, the square root of, 5 squared minus 4, times 6, times 7, end root, and denominator 2 times 6, end fraction, or the fraction with numerator negative 5 plus or minus, the square root of, 25 minus 168, end root, and denominator 12, end fraction.

Choice B is incorrect and may result from using the fraction with numerator negative a, plus or minus, the square root of, b squared, minus 4 a, c, end root, and denominator 2 a, end fraction as the quadratic formula. Choice C is incorrect and may result from usingthe fraction with numerator negative b plus or minus, the square root of, a, squared, plus 4 a, c, end root, and denominator 2 a, end fraction as the quadratic formula. Choice D is incorrect and may result from using the fraction with numerator negative a, plus or minus, the square root of, a, squared, plus 4 a, c, end root, and denominator 2 a, end fraction as the quadratic formula.

 

Question 380 380 of 479 selected Equivalent Expressions M

f(x)=x2+bx

g(x)=9x2-27x

Functions f and g are given, and in function f , b is a constant. If f(x)·g(x)=9x4-26x3-3x2, what is the value of b ?

  1. -26

  2. - 26 9

  3. 1 9

  4. 9

Show Answer Correct Answer: C

Choice C is correct. Multiplying the given functions f and g yields f(x)·g(x)=(x2+bx)(9x2-27x). Applying the distributive property to the right-hand side of this equation yields f(x)·g(x)=(x2)(9x2-27x)+(bx)(9x2-27x). Applying the distributive property once again to the right-hand side of this equation yields f(x)·g(x)=(x2)(9x2)-(x2)(27x)+(bx)(9x2)-(bx)(27x), which is equivalent to f(x)·g(x)=9x4-27x3+9bx3-27bx2. Factoring out x3 from the second and third terms yields f(x)·g(x)=9x4+(-27+9b)x3-27bx2. Since the left-hand sides of f(x)·g(x)=9x4+(-27+9b)x3-27bx2 and f(x)·g(x)=9x4-26x3-3x2 are equal, it follows that (-27+9b)x3=-26x3, or -27+9b=-26, and -27bx2=-3x2, or -27b=-3. Adding 27 to each side of -27+9b=-26 yields 9b=1. Dividing each side of this equation by 9 yields b=19. Similarly, dividing each side of -27b=-3 by - 27 yields b=19. Therefore, the value of b is 19.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 381 381 of 479 selected Nonlinear Functions M

The equation E(t)=5(1.8)t gives the estimated number of employees at a restaurant, where t is the number of years since the restaurant opened. Which of the following is the best interpretation of the number 5 in this context?

  1. The estimated number of employees when the restaurant opened

  2. The increase in the estimated number of employees each year

  3. The number of years the restaurant has been open

  4. The percent increase in the estimated number of employees each year

Show Answer Correct Answer: A

Choice A is correct. For an exponential function of the form E(t)=a(b)t, where a and b are constants, the initial value of the function—that is, the value of the function when t = 0 —is a and the value of the function increases by a factor of b each time t increases by 1 . Since the function E(t)=5(1.8)t gives the estimated number of employees at a restaurant and t is the number of years since the restaurant opened, the best interpretation of the number 5 in this context is the estimated number of employees when t = 0 , or when the restaurant opened.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 382 382 of 479 selected Nonlinear Functions E

The kinetic energy, in joules, of an object with mass 9 kilograms traveling at a speed of v meters per second is given by the function K , where K(v)=92v2. Which of the following is the best interpretation of K(34)=5,202 in this context?

  1. The object traveling at 34 meters per second has a kinetic energy of 5,202 joules.

  2. The object traveling at 340 meters per second has a kinetic energy of 5,202 joules.

  3. The object traveling at 5,202 meters per second has a kinetic energy of 34 joules.

  4. The object traveling at 23,409 meters per second has a kinetic energy of 34 joules.

Show Answer Correct Answer: A

Choice A is correct. It's given that the kinetic energy, in joules, of an object with a mass of 9 kilograms traveling at a speed of v meters per second is given by the function K , where K(v)=92v2. It follows that in the equation K(34)=5,202, 34 is the value of v , or the speed of the object, in meters per second, and 5,202 is the kinetic energy, in joules, of the object at that speed. Therefore, the best interpretation of K(34)=5,202 in this context is the object traveling at 34 meters per second has a kinetic energy of 5,202 joules.

Choice B is incorrect. The object traveling at 340 meters per second has a kinetic energy of 520,200 joules.

Choice C is incorrect. The object traveling at 5,202 meters per second has a kinetic energy of 121,773,618 joules.

Choice D is incorrect. The object traveling at 23,409 meters per second has a kinetic energy of 2,465,915,764.5 joules.

Question 383 383 of 479 selected Equivalent Expressions M

open parenthesis, one-half x, plus 3, close parenthesis, minus, open parenthesis, two-thirds x, minus 5, close parenthesis

Which of the following is equivalent to the expression above?

  1. negative one sixth x, plus 8
  2. negative one sixth x, minus 2
  3. negative one-third x squared, plus one-half x, plus 15
  4. negative one-third x squared, minus nine halves x, minus 15
Show Answer Correct Answer: A

Choice A is correct. By distributing the minus sign through the expression two thirds x minus 5, the given expression can be rewritten as open parenthesis, one half x plus 3, close parenthesis, minus two thirds x, plus 5, which is equivalent to one half x, minus two thirds x, plus 3, plus 5. Combining like terms gives open parenthesis, one half minus two thirds, close parenthesis, times x, plus, open parenthesis, 3 plus 5, close parenthesis, or negative one sixth x plus 8.

Choice B is incorrect and may be the result of failing to distribute the minus sign appropriately through the second term and simplifying the expression one half x, plus 3, minus two thirds x, minus 5. Choice C is incorrect and may be the result of multiplying the expressions one half x plus 3 and negative two thirds x plus 5. Choice D is incorrect and may be the result of multiplying the expressions one half x plus 3 and negative two thirds x minus 5.

Question 384 384 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

the fraction with numerator 4 x squared, and denominator x squared minus 9, end fraction, minus the fraction with numerator 2 x, and denominator x plus 3, end fraction, equals the fraction with numerator 1, and denominator x minus 3, end fraction

What value of x satisfies the equation above?

  1. negative 3

  2. negative one half

  3. one half

  4. 3

Show Answer Correct Answer: C

Choice C is correct. Each fraction in the given equation can be expressed with the common denominator x squared, minus 9. Multiplying the fraction with numerator 2 x, and denominator x plus 3, end fraction by the fraction with numerator x minus 3, and denominator x minus 3, end fraction yields the fraction with numerator 2 x squared, minus 6, and denominator x squared minus 9, end fraction , and multiplying the fraction with numerator 1, and denominator x minus 3, end fraction by the fraction with numerator x plus 3, and denominator x plus 3, end fraction yields the fraction with numerator x plus 3, and denominator x squared minus 9, end fraction. Therefore, the given equation can be written as the fraction with numerator 4 x squared, and denominator x squared minus 9, minus, the fraction with numerator 2 x squared, minus 6 x, and denominator x squared minus 9, end fraction, equals, the fraction with numerator x plus 3, and denominator x squared minus 9, end fraction. Multiplying each fraction by the denominator results in the equation 4 x squared minus, open parenthesis, 2 x squared, minus 6 x, close parenthesis, equals, x plus 3, or 2 x squared, plus 6 x, equals, x plus 3. This equation can be solved by setting a quadratic expression equal to 0, then solving for x. Subtracting x plus 3 from both sides of this equation yields 2 x squared, plus 5 x, minus 3, equals 0. The expression 2 x squared, plus 5 x, minus 3 can be factored, resulting in the equation open parenthesis, 2 x minus 1, close parenthesis, times, open parenthesis, x plus 3, close parenthesis, equals 0. By the zero product property, 2 x minus 1, equals 0 or x plus 3, equals 0. To solve for x in 2 x minus 1, equals 0, 1 can be added to both sides of the equation, resulting in 2 x equals 1. Dividing both sides of this equation by 2 results in x equals one half. Solving for x in x plus 3, equals 0 yields x equals negative 3. However, this value of x would result in the second fraction of the original equation having a denominator of 0. Therefore, x equals negative 3 is an extraneous solution. Thus, the only value of x that satisfies the given equation is x equals one half.

Choice A is incorrect and may result from solving x plus 3, equals 0 but not realizing that this solution is extraneous because it would result in a denominator of 0 in the second fraction. Choice B is incorrect and may result from a sign error when solving 2 x minus 1, equals 0 for x. Choice D is incorrect and may result from a calculation error.

 

Question 385 385 of 479 selected Equivalent Expressions M

open parenthesis, 4, x cubed, minus 5, x squared, plus 3, close parenthesis, minus, open parenthesis, 6, x cubed, plus 2, x squared, minus x, close parenthesis

Which of the following expressions is equivalent to the expression above?

  1. negative 10, x cubed, minus 3, x squared, plus x, plus 3

  2. negative 2, x cubed, minus 7, x squared, plus x, plus 3

  3. negative 2, x cubed, minus 3, x squared, plus x, plus 3

  4. 10, x cubed, minus 7, x squared, minus x, plus 3

Show Answer Correct Answer: B

Choice B is correct. Using the distributive property, the given expression can be rewritten as 4 x cubed, minus 5 x squared, plus 3, minus 6 x cubed, minus 2 x squared, plus x. Combining like terms, this expression can be rewritten as open parenthesis, 4 minus 6, close parenthesis, times x cubed, plus, open parenthesis, negative 5 minus 2, close parenthesis, times x squared, plus x, plus 3, which is equivalent to negative 2 x cubed, minus 7 x squared, plus x, plus 3.

Choices A, C, and D are incorrect and may result from an error when applying the distributive property or an error when combining like terms.

 

Question 386 386 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

x squared, equals 64

Which of the following values of x satisfies the given equation?

  1. negative 8

  2. 4

  3. 32

  4. 128

Show Answer Correct Answer: A

Choice A is correct. Solving for x by taking the square root of both sides of the given equation yields x equals 8 or x equals negative 8. Of the choices given, negative 8 satisfies the given equation.

Choice B is incorrect and may result from a calculation error when solving for x. Choice C is incorrect and may result from dividing 64 by 2 instead of taking the square root. Choice D is incorrect and may result from multiplying 64 by 2 instead of taking the square root.

 

Question 387 387 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

7 x 2 - 20 x - 32 = 0

What is the positive solution to the given equation?

Show Answer Correct Answer: 4

The correct answer is 4 . The left-hand side of the given equation can be factored as (7x+8)(x-4). Therefore, the given equation, 7x2-20x-32=0, can be written as (7x+8)(x-4)=0. Applying the zero product property to this equation yields 7 x + 8 = 0 and x - 4 = 0 . Subtracting 8 from both sides of the equation 7 x + 8 = 0 yields 7 x = - 8 . Dividing both sides of this equation by 7 yields x = - 8 7 . Adding 4 to both sides of the equation x - 4 = 0 yields x = 4 . Therefore, the two solutions to the given equation, 7 x 2 - 20 x - 32 = 0 , are - 8 7 and 4 . It follows that 4 is the positive solution to the given equation.

Question 388 388 of 479 selected Nonlinear Functions H

f(x)= 4 x 2 - 50 x + 126

The given equation defines the function f . For what value of x  does f(x) reach its minimum?

Show Answer Correct Answer: 25/4, 6.25

The correct answer is 25 4 . The given equation can be rewritten in the form f(x)=a(x-h)2+k, where a , h , and k are constants. When a>0, h is the value of x for which f(x) reaches its minimum. The given equation can be rewritten as f(x)=4(x2-504x)+126, which is equivalent to f(x)=4(x2-504x+(508)2-(508)2)+126. This equation can be rewritten as f(x)=4((x-508)2-(508)2)+126, or f(x)=4(x-508)2-4(508)2+126, which is equivalent to f(x)=4(x-254)2-1214. Therefore, h = 25 4 , so the value of x for which f(x) reaches its minimum is 25 4 . Note that 25/4 and 6.25 are examples of ways to enter a correct answer.

Question 389 389 of 479 selected Nonlinear Functions M

The function f is defined by f(x)=|x-4x|. What value of a satisfies f(5)-f(a)=-15?

  1. -20

  2. 5

  3. 10

  4. 45

Show Answer Correct Answer: C

Choice C is correct. It's given that the function f is defined by f(x)=|x-4x|. It's also given that f(5)-f(a)=-15. Substituting 5 for x in the function f(x)=|x-4x| yields f(5)=|5-4(5)| and substituting a for x in the function f(x)=|x-4x| yields f(a)=|a-4a|. Therefore, f(5)=15 and f(a)=|-3a|. Substituting 15 for f(5) and |-3a| for f(a) in the equation f(5)-f(a)=-15 yields 15-|-3a|=-15. Subtracting 15 from both sides of this equation yields -|-3a|=-30. Dividing both sides of this equation by - 1 yields |-3a|=30. By the definition of absolute value, if |-3a|=30, then - 3 a = 30 or - 3 a = - 30 . Dividing both sides of each of these equations by - 3 yields a = - 10 or a = 10 , respectively. Thus, of the given choices, a value of a that satisfies f(5)-f(a)=-15 is 10 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 390 390 of 479 selected Nonlinear Functions H

h of x equals, x cubed plus, a, x squared, plus b x, plus c

The function h is defined above, where a, b, and c are integer constants. If the zeros of the function are negative 5 6, and 7, what is the value of c ?

Show Answer

The correct answer is 210. Since negative 5, 6, and 7 are zeros of the function, the function can be rewritten as h of x equals, open parenthesis, x plus 5, close parenthesis, times, open parenthesis, x minus 6, close parenthesis, times, open parenthesis, x minus 7, close parenthesis. Expanding the function yields h of x equals, x cubed, minus 8, x squared, minus 23 x, plus 210. Thus, a, equals negative 8, b equals negative 23, and c equals 210. Therefore, the value of c is 210.

Question 391 391 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

  • For the absolute value function:
    • Moving from left to right:
      • The function slants sharply down to the point (4 comma 2).
      • The function then slants sharply up.
    • The function passes through the following points:
      • (1 comma 5) 
      • (4 comma 2)
      • (6 comma 4)
  • For the linear function:
    • The function slants sharply up from left to right.
    • The function passes through the following points:
      • (0 comma 4)
      • (1 comma 5)

The graph of a system of an absolute value function and a linear function is shown. What is the solution (x,y) to this system of two equations?

  1. (-1,5)

  2. (0,4)

  3. (1,5)

  4. (4,2)

Show Answer Correct Answer: C

Choice C is correct. The solution to the system of two equations corresponds to the point where the graphs of the equations intersect. The graphs of the linear function and the absolute value function shown intersect at the point (1,5). Thus, the solution to the system is (1,5).

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the y-intercept of the graph of the linear function.

Choice D is incorrect. This is the vertex of the graph of the absolute value function.

Question 392 392 of 479 selected Nonlinear Functions E

The function f is defined by f(x)=8x. For what value of x does f(x)=48?

  1. 6

  2. 8

  3. 36

  4. 64

Show Answer Correct Answer: C

Choice C is correct. It's given that f(x)=8x. Substituting 48 for f(x) in this equation yields 48=8x. Dividing both sides of this equation by 8 yields 6=x. This can be rewritten as x=6. Squaring both sides of this equation yields x=36. Therefore, the value of x for which f(x)=48 is 36 .

Choice A is incorrect. If x=6, f(x)=86, not 48 .

Choice B is incorrect. If x=8, f(x)=88, not 48 .

Choice D is incorrect. If x=64, f(x)=864, which is equivalent to 64 , not 48 .

Question 393 393 of 479 selected Equivalent Expressions M

Which of the following is equivalent to the fourth root of, x squared, plus 8 x, plus 16, end root, where x is greater than 0?

  1. open parenthesis, x plus 4, close parenthesis, to the fourth power

  2. open parenthesis, x plus 4, close parenthesis, squared

  3. open parenthesis, x plus 4, close parenthesis

  4. open parenthesis, x plus 4, close parenthesis, raised to the one half power

Show Answer Correct Answer: D

Choice D is correct. The given expression can also be written as open parenthesis, x squared, plus 8, x, plus 16, close parenthesis, raised to the one fourth power. The trinomial x squared, plus 8, x, plus 16 can be rewritten in factored form as open parenthesis, x plus 4, close parenthesis, squared. Thus, the entire expression can be rewritten as open parenthesis, open parenthesis, x plus 4, close parenthesis, squared, close parenthesis, raised to the one fourth power. Simplifying the exponents yields open parenthesis, x plus 4, close parenthesis, raised to the one half power.

Choices A, B, and C are incorrect and may result from errors made when simplifying the exponents in the expression open parenthesis, open parenthesis, x plus 4, close parenthesis, squared, close parenthesis, raised to the one fourth power.

 

Question 394 394 of 479 selected Nonlinear Functions H

f(x)= ( x - 2 ) ( x + 15 )

The function f is defined by the given equation. For what value of x does f(x) reach its minimum?

Show Answer Correct Answer: -13/2, -6.5

The correct answer is -132. The value of x for which f(x) reaches its minimum can be found by rewriting the given equation in the form f(x)=(x-h)2+k, where f(x) reaches its minimum, k , when the value of x is h . The given equation, f(x)=(x-2)(x+15), can be rewritten as f(x)=x2+13x-30. By completing the square, this equation can be rewritten as f(x)=(x2+13x+(132)2)-30-(132)2, which is equivalent to f(x)=(x+132)2-2894, or f(x)=(x-(-132))2-2894. Therefore, f(x) reaches its minimum when the value of x is -132. Note that -13/2 and -6.5 are examples of ways to enter a correct answer.

Alternate approach: The graph of y=f(x) in the xy-plane is a parabola. The value of x for the vertex of a parabola is the x-value of the midpoint between the two x-intercepts of the parabola. Since it's given that f(x)=(x-2)(x+15), it follows that the two x-intercepts of the graph of y=f(x) in the xy-plane occur when x = 2 and x = - 15 , or at the points (2,0) and (-15,0). The midpoint between two points, (x1,y1) and (x2,y2), is (x1+x22,y1+y22). Therefore, the midpoint between (2,0) and (-15,0) is (2-152,0+02), or (-132,0). It follows that f(x) reaches its minimum when the value of x is -132. Note that -13/2 and -6.5 are examples of ways to enter a correct answer.

Question 395 395 of 479 selected Nonlinear Functions E

  • For the first curve:
    • Moving from left to right:
      • The curve is in quadrant 3.
      • As x decreases, the curve approaches the line y equals 0.
      • As x increases, the curve approaches the line x equals negative 4.
  • For the second curve:
    • Moving from left to right:
      • The curve passes from quadrant 2 to quadrant 1.
      • As x decreases, the curve approaches the line x equals negative 4.
      • As x increases, the curve approaches the line y equals 0.
  • The 2 curves pass through the following points:
    • approximately (negative 7 comma negative 4.9)
    • (0 comma 3)
    • approximately (7 comma 0.8)

The graph of y=f(x) is shown in the xy-plane. The value of f(0) is an integer. What is the value of f(0)?

Show Answer Correct Answer: 3

The correct answer is 3. The value of f(0) is the value of y on the graph of y=f(x) in the xy-plane that corresponds with x=0. It's given that the value of f(0) is an integer. For the graph of y=f(x) shown, when x=0, the corresponding integer value of y is 3. Therefore, the value of f(0) is 3.

Question 396 396 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

64 x 2 + b x + 25 = 0

In the given equation, b is a constant. For which of the following values of b will the equation have more than one real solution?

  1. -91

  2. -80

  3. 5

  4. 40

Show Answer Correct Answer: A

Choice A is correct. A quadratic equation of the form ax2+bx+c=0, where a , b , and c are constants, has either no real solutions, exactly one real solution, or exactly two real solutions. That is, for the given equation to have more than one real solution, it must have exactly two real solutions. When the value of the discriminant, or b2-4ac, is greater than 0, the given equation has exactly two real solutions. In the given equation, 64x2+bx+25=0a=64 and c=25. Therefore, the given equation has exactly two real solutions when (b)2-4(64)(25)>0, or b2-6,400>0. Adding 6,400 to both sides of this inequality yields b2>6,400. Taking the square root of both sides of b2>6,400 yields two possible inequalities: b<-80 or b>80. Of the choices, only choice A satisfies b<-80 or b>80.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 397 397 of 479 selected Nonlinear Functions H

f(x)=9(4)x

The function f is defined by the given equation. If g(x)=f(x+2), which of the following equations defines the function g ?

  1. g(x)=18(4)x

  2. g(x)=144(4)x

  3. g(x)=18(8)x

  4. g(x)=81(16)x

Show Answer Correct Answer: B

Choice B is correct. It’s given that f(x)=9(4)x and g(x)=f(x+2). Substituting x+2 for x in f(x)=9(4)x gives f(x+2)=9(4)x+2. Rewriting this equation using properties of exponents gives f(x+2)=9(4)x(4)2, which is equivalent to f(x+2)=9(4)x(16). Multiplying 9 and 16 in this equation gives f(x+2)=144(4)x. Since g(x)=f(x+2), g(x)=144(4)x.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 398 398 of 479 selected Equivalent Expressions H

If a and c are positive numbers, which of the following is equivalent to the square root of, open parenthesis, a plus c, close parenthesis, cubed, end root, times, the square root of a plus c, end root?

  1. a plus c
  2. a squared plus c squared
  3. a squared plus 2 a c plus c squared
  4. a squared times c squared
Show Answer Correct Answer: C

Choice C is correct. Using the property that the square root of x, end root, times the square root of y, end root, equals the square root of x y, end root for positive numbers x and y, with x = (a + c)3 and y = a + c, it follows that the square root of, open parenthesis, a, plus c, close parenthesis, cubed, end root, times, the square root of a, plus c, end root, equals the square root of, open parenthesis a, plus c, end parenthesis to the power 4, end root. By rewriting (a + c)4 as ((a + c)2)2, it is possible to simplify the square root expression as follows: the square root of, open outer parenthesis, open inner parenthesis, a, plus c, close inner parenthesis, squared, close outer parenthesis, squared, end root, equals, open parenthesis, a, plus c, close parenthesis, squared, which equals a, squared, plus 2 a, c, plus c squared.

Choice A is incorrect and may be the result of the square root of, open parenthesis, a, plus c, close parenthesis, cubed, end root, divided by the square root of,  a, plus c, end root. Choice B is incorrect and may be the result of incorrectly rewriting (a + c)2 as a2 + c2. Choice D is incorrect and may be the result of incorrectly applying properties of exponents.

Question 399 399 of 479 selected Nonlinear Functions H

The functions f and g are defined by the given equations, where x0. Which of the following equations displays, as a constant or coefficient, the maximum value of the function it defines, where x0?

  1. f(x)=18(1.25)x+41
  2. g(x)=9(0.73)x
  1. I only

  2. II only

  3. I and II

  4. Neither I nor II

Show Answer Correct Answer: B

Choice B is correct. For the function f , since the base of the exponent, 1.25 , is greater than 1 , the value of (1.25)x increases as x increases. Therefore, the value of 18(1.25)x and the value of 18(1.25)x+41 also increase as x increases. Since f is therefore an increasing function where x0, the function f has no maximum value. For the function g , since the base of the exponent, 0.73 , is less than 1 , the value of (0.73)x decreases as x increases. Therefore, the value of 9(0.73)x also decreases as x increases. It follows that the maximum value of g(x) for x0 occurs when x = 0 . Substituting 0 for x in the function g yields g(0)=9(0.73)0, which is equivalent to g(0)=9(1), or g(0)=9. Therefore, the maximum value of g(x) for x0 is 9 , which appears as a coefficient in equation II. So, of the two equations given, only II displays, as a constant or coefficient, the maximum value of the function it defines, where x0.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 400 400 of 479 selected Nonlinear Functions H

A rectangle has an area of 155 square inches. The length of the rectangle is 4 inches less than 7 times the width of the rectangle. What is the width of the rectangle, in inches?

Show Answer Correct Answer: 5

The correct answer is 5 . Let x represent the width, in inches, of the rectangle. It's given that the length of the rectangle is 4 inches less than 7 times its width, or 7 x - 4 inches. The area of a rectangle is equal to its width multiplied by its length. Multiplying the width, x inches, by the length, 7 x - 4 inches, yields x(7x-4) square inches. It’s given that the rectangle has an area of 155 square inches, so it follows that x(7x-4)=155, or 7x2-4x=155. Subtracting 155 from both sides of this equation yields 7 x 2 - 4 x - 155 = 0 . Factoring the left-hand side of this equation yields (7x+31)(x-5)=0. Applying the zero product property to this equation yields two solutions: x=-317 and x = 5 . Since x is the rectangle’s width, in inches, which must be positive, the value of x is 5 . Therefore, the width of the rectangle, in inches, is 5 .

Question 401 401 of 479 selected Nonlinear Functions H

f(x)= ( x + 7 ) 2 + 4

The function f is defined by the given equation. For what value of x does f(x) reach its minimum?

Show Answer Correct Answer: -7

The correct answer is -7 . For a quadratic function defined by an equation of the form  f(x)=a(x-h)2+k, where a , h , and k are constants and a>0, the function reaches its minimum when x = h . In the given function, a = 1 , h = -7 , and k = 4 . Therefore, the value of x for which f(x) reaches its minimum is -7 .

Question 402 402 of 479 selected Nonlinear Functions M

The product of a positive number x and the number that is 8 more than x is 180 . What is the value of x ?

  1. 5

  2. 10

  3. 18

  4. 36

Show Answer Correct Answer: B

Choice B is correct. The number that's 8 more than x can be represented by the expression x + 8 . It's given that the product of x and x + 8 is 180 , so it follows that (x)(x+8)=180, or x 2 + 8 x = 180 . Subtracting 180 from each side of this equation yields x 2 + 8 x - 180 = 0 . Factoring the left-hand side of this equation yields (x-10)(x+18)=0. Applying the zero product property to this equation yields two solutions: x = 10 and x = - 18 . Since x is a positive number, the value of x is 10 .

Choice A is incorrect. If x = 5 , the product of x and the number that's 8 more than x would be (5)(13), or 65 , not 180 .

Choice C is incorrect. This is the value of the number that's 8 more than x , not the value of x .

Choice D is incorrect. If x = 36 , the product of x and the number that's 8 more than x would be (36)(44), or 1,584 , not 180 .

Question 403 403 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

y = x 2 + 14 x + 48

x + 8 = 11

The solution to the given system of equations is (x,y). What is the value of y ?

Show Answer Correct Answer: 99

The correct answer is 99 . In the given system of equations, the second equation is x+8=11. Subtracting 8 from both sides of this equation yields x=3. In the given system of equations, the first equation is y=x2+14x+48. Substituting 3 for x in this equation yields y=(3)2+14(3)+48, or y=99. Therefore, the solution to the given system of equations is (x,y)=(3,99). Thus, the value of y is 99 .

Question 404 404 of 479 selected Equivalent Expressions E

Which expression is equivalent to 15w2+8w?

  1. w(15w+8)

  2. 8w(15w+1)

  3. 15w2(8w+1)

  4. 23(w2+w)

Show Answer Correct Answer: A

Choice A is correct. Since each term of the given expression has a common factor of w, it may be rewritten as w(15w+8). Therefore, the expression w(15w+8) is equivalent to 15w2+8w.

Choice B is incorrect. This expression is equivalent to 120w2+8w, not 15w2+8w.

Choice C is incorrect. This expression is equivalent to 120w3+15w2, not 15w2+8w.

Choice D is incorrect. This expression is equivalent to 23w2+23w, not 15w2+8w.

Question 405 405 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

The total revenue from sales of a product can be calculated using the formula T equals, P Q, where T is the total revenue, P is the price of the product, and Q is the quantity of the product sold. Which of the following equations gives the quantity of product sold in terms of P and T ?

  1. Q equals, P over T

  2. Q equals, T over P

  3. Q equals, P T

  4. Q equals, T minus P

Show Answer Correct Answer: B

Choice B is correct. Solving the given equation for Q gives the quantity of the product sold in terms of P and T. Dividing both sides of the given equation by P yields T over P, equals Q, or Q equals, T over P. Therefore, Q equals, T over P gives the quantity of product sold in terms of P and T.

Choice A is incorrect and may result from an error when dividing both sides of the given equation by P. Choice C is incorrect and may result from multiplying, rather than dividing, both sides of the given equation by P. Choice D is incorrect and may result from subtracting P from both sides of the equation rather than dividing both sides by P.

 

Question 406 406 of 479 selected Nonlinear Functions E

The function f is defined by f(x)=x3+15. What is the value of f(2)?

  1. 20

  2. 21

  3. 23

  4. 24

Show Answer Correct Answer: C

Choice C is correct. The value of f(2) is the value of f(x) when x = 2 . Substituting 2 for x in the given function yields f(2)=(2)3+15, or f(2)=8+15, which is equivalent to f(2)=23. Therefore, the value of f(2) is 23 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the value of f(2) when f(x)=x(3)+15, rather than f(x)=x3+15.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 407 407 of 479 selected Nonlinear Functions H

f(x)=272(2)x

The function f is defined by the given equation. If h(x)=f(x-4), which of the following equations defines function h ?

  1. h(x)=17(2)x

  2. h(x)=68(2)x

  3. h(x)=272(16)x

  4. h(x)=272(8)x

Show Answer Correct Answer: A

Choice A is correct. It's given that f(x)=272(2)x and h(x)=f(x-4). Substituting x - 4 for x in f(x)=272(2)x yields f(x-4)=272(2)x-4. Substituting h(x) for f(x-4) in this equation yields h(x)=272(2)x-4. Using the properties of exponents, the function h(x)=272(2)x-4 can be rewritten as h(x)=272(2)x24, which is equivalent to h(x)=272(2)x16, or h(x)=17(2)x. Therefore, of the given choices, an equation that defines function h is h(x)=17(2)x.

Choice B is incorrect. This equation defines function h if h(x)=f(x-2), not h(x)=f(x-4).

Choice C is incorrect. This equation defines function h if h(x)=f(4x), not h(x)=f(x-4).

Choice D is incorrect and may result from conceptual or calculation errors.

Question 408 408 of 479 selected Nonlinear Functions E

  • Moving from left to right, the curve passes from quadrant 2 to quadrant 1.
  • The curve has 1 relative maximum and 1 relative minimum.
  • The curve passes through the following points:
    • (0 comma 40)
    • (4 comma 40)
    • (7 comma 40)

The y-intercept of the graph shown is (x,y). What is the value of y ?

Show Answer Correct Answer: 40

The correct answer is 40 . The y-intercept of a graph in the xy-plane is the point (x,y) on the graph where x=0. The y-intercept of the graph shown is (0,40). Therefore, the value of y is 40 .

Question 409 409 of 479 selected Nonlinear Functions H

Kao measured the temperature of a cup of hot chocolate placed in a room with a constant temperature of 70 degrees Fahrenheit (°F). The temperature of the hot chocolate was 185°F at 6:00 p.m. when it started cooling. The temperature of the hot chocolate was 156°F at 6:05 p.m. and 135°F at 6:10 p.m. The hot chocolate’s temperature continued to decrease. Of the following functions, which best models the temperature T of m, in degrees Fahrenheit, of Kao’s hot chocolate m minutes after it started cooling?

  1. T of m equals, 185 times, 1 point 2 5 to the m power

  2. T of m equals, 185 times, 0 point 8 5 to the m power

  3. T of m equals, open parenthesis, 185 minus 70, close parenthesis, times, 0 point 7 5 raised to the m over 5 power

  4. T of m equals, 70 plus, 115 times, 0 point 7 5 raised to the m over 5 power

Show Answer Correct Answer: D

Choice D is correct. The hot chocolate cools from 185°F over time, never going lower than the room temperature, 70°F. Since the base of the exponent in this function, 0.75, is less than 1, T of m decreases as time increases. Using the function, the temperature, in °F, at 6:00 p.m. can be estimated as T of 0 and is equal to 70 plus 115, times, open parenthesis, 0 point 7 5, close parenthesis, raised to the power zero fifths, equals 185. The temperature, in °F, at 6:05 p.m. can be estimated as T of 5 and is equal to 70 plus 115, times, open parenthesis, 0 point 7 5, close parenthesis raised to the power five fifths, which is approximately 156°F. Finally, the temperature, in °F, at 6:10 p.m. can be estimated as T of 10 and is equal to 70 plus 115, times, open parenthesis, 0 point 7 5, close parenthesis, raised to the power ten fifths, which is approximately 135°F. Since these three given values of m and their corresponding values for T of m can be verified using the function T of m equals, 70 plus 115, times, open parenthesis, 0 point 7 5, close parenthesis, raised to the power the fraction m over 5, this is the best function out of the given choices to model the temperature of Kao’s hot chocolate after m minutes.

Choice A is incorrect because the base of the exponent,1 point 2 5, results in the value of T of m increasing over time rather than decreasing. Choice B is incorrect because when m is large enough, T of m becomes less than 70. Choice C is incorrect because the maximum value of T of m at 6:00 p.m. is 115°F, not 185°F.

 

Question 410 410 of 479 selected Nonlinear Functions E

  • In quadrant 3, the curve rises to its maximum at point (0 comma negative 3).
  • In quadrant 4, the curve then falls.

The graph of the polynomial function f , where y=f(x), is shown. The y-intercept of the graph is (0,y). What is the value of y ?

Show Answer Correct Answer: -3

The correct answer is -3. The y-intercept of the graph of a function in the xy-plane is the point where the graph crosses the y-axis. The graph of the polynomial function shown crosses the y-axis at the point (0,-3). It's given that the y-intercept of the graph is (0,y). Thus, the value of y is -3.

Question 411 411 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

y = -1.5

y = x 2 + 8 x + a

In the given system of equations, a is a positive constant. The system has exactly one distinct real solution. What is the value of a ?

Show Answer Correct Answer: 14.5, 29/2

The correct answer is 292. According to the first equation in the given system, the value of y is -1.5 . Substituting -1.5 for y in the second equation in the given system yields -1.5=x2+8x+a. Adding 1.5 to both sides of this equation yields 0=x2+8x+a+1.5. If the given system has exactly one distinct real solution, it follows that 0=x2+8x+a+1.5 has exactly one distinct real solution. A quadratic equation in the form 0=px2+qx+r, where p , q , and r are constants, has exactly one distinct real solution if and only if the discriminant, q2-4pr, is equal to 0 . The equation 0=x2+8x+a+1.5 is in this form, where p=1, q=8, and r=a+1.5. Therefore, the discriminant of the equation 0=x2+8x+a+1.5 is (8)2-4(1)(a+1.5), or 58-4a. Setting the discriminant equal to 0 to solve for a yields 58-4a=0. Adding 4 a to both sides of this equation yields 58=4a. Dividing both sides of this equation by 4 yields 584=a, or 292=a. Therefore, if the given system of equations has exactly one distinct real solution, the value of a is 292. Note that 29/2 and 14.5 are examples of ways to enter a correct answer.

Question 412 412 of 479 selected Nonlinear Functions M

f of x equals, open parenthesis, x plus 4, close parenthesis, times, open parenthesis, x minus 1, close parenthesis, times, open parenthesis, 2 x minus 3, close parenthesis

The function f is defined above. Which of the following is NOT an x-intercept of the graph of the function in the xy-plane?

  1. the point with coordinates negative 4 comma 0

  2. the point with coordinates negative two thirds comma 0

  3. the point with coordinates 1 comma 0

  4. the point with coordinates three halves comma 0

Show Answer Correct Answer: B

Choice B is correct. The graph of the function f in the xy-plane has x-intercepts at the points with coordinates x comma y, where y equals f of x, which equals 0. Substituting 0 for f of x in the given equation yields 0 equals, open parenthesis, x plus 4, close parenthesis, times, open parenthesis, x minus 1, close parenthesis, times, open parenthesis, 2 x minus 3, close parenthesis. By the zero product property, if 0 equals, open parenthesis, x plus 4, close parenthesis, times, open parenthesis, x minus 1, close parenthesis, times, open parenthesis, 2 x minus 3, close parenthesis, then x plus 4, equals 0, x minus 1, equals 0, or 2 x minus 3, equals 0. Solving each of these linear equations for x, it follows that x equals negative 4, x equals 1, and x equals three halves, respectively. This means that the graph of the function f in the xy-plane has three x-intercepts: the point with coordinates negative 4 comma 0, the point with coordinates 1 comma 0, and the point with coordinates three halves comma 0. Therefore, the point with coordinates negative two thirds comma 0 isn’t an x-intercept of the graph of the function f.
Alternate approach: Substitution may be used. Since by definition an x-intercept of any graph is a point in the form k comma 0 where k is a constant, and since all points in the options are in this form, it need only be checked whether the points in the options lie on the graph of the function f. Substituting negative two thirds for x and 0 for f of x in the given equation yields 0 equals, open parenthesis, negative two thirds plus 4, close parenthesis, times, open parenthesis, negative two thirds minus 1, close parenthesis, times, open parenthesis, 2 times negative two thirds, minus 3, close parenthesis, or 0 equals, the fraction 650 over 27. Therefore, the point with coordinates negative two thirds comma 0 doesn’t lie on the graph of the function f and can’t be an x-intercept of the graph.

Choices A, C, and D are incorrect because each of these points is an x-intercept of the graph of the function f in the xy-plane. By definition, an x-intercept is a point on the graph of the form k comma 0 , where k is a constant. Substituting negative 4 for x and 0 for f of x in the given equation yields 0 equals, open parenthesis, negative 4 plus 4, close parenthesis, times, open parenthesis, negative 4 minus 1, close parenthesis, times, open parenthesis, 2 times negative 4, minus 3, close parenthesis, or 0 equals 0. Since this is a true statement, the point with coordinates negative 4 comma 0 lies on the graph of the function f and is an x-intercept of the graph. Performing similar substitution using the points with coordinates 1 comma 0 and three halves comma 0 also yields the true statement 0 equals 0, illustrating that these points also lie on the graph of the function f and are x-intercepts of the graph.

 

Question 413 413 of 479 selected Equivalent Expressions E

Which expression is equivalent to 8+d2+3?

  1. d 2 + 24

  2. d 2 + 11

  3. d 2 + 5

  4. d 2 - 11

Show Answer Correct Answer: B

Choice B is correct. The given expression can be rewritten as d2+8+3. Adding 8 and 3 in this expression yields  d 2 + 11 .

Choice A is incorrect. This expression is equivalent to d2+8(3).

Choice C is incorrect. This expression is equivalent to 8+d2-3.

Choice D is incorrect. This expression is equivalent to -8+d2-3.

Question 414 414 of 479 selected Nonlinear Functions E

  • Moving from left to right:
    • The curve passes from quadrant 2 to quadrant 1.
    • In quadrant 2, the curve trends down sharply to the point (0 comma 3).
    • In quadrant 1, the curve trends down gradually.
  • As x increases, the curve approaches the line y equals 0.
  • The curve passes through the following points:
    • (0 comma 3)
    • approximately (0.5 comma 0.7)

The graph of the exponential function f is shown, where y=f(x). The y-intercept of the graph is (0,y). What is the value of y ?

Show Answer Correct Answer: 3

The correct answer is 3 . It's given that the y-intercept of the graph shown is (0,y). The graph passes through the point (0,3). Therefore, the value of y is 3 .

Question 415 415 of 479 selected Nonlinear Functions H

The quadratic function g models the depth, in meters, below the surface of the water of a seal t minutes after the seal entered the water during a dive. The function estimates that the seal reached its maximum depth of 302.4 meters 6 minutes after it entered the water and then reached the surface of the water 12 minutes after it entered the water. Based on the function, what was the estimated depth, to the nearest meter, of the seal 10 minutes after it entered the water?

Show Answer Correct Answer: 168

The correct answer is 168 . The quadratic function g gives the estimated depth of the seal, g(t), in meters, t minutes after the seal enters the water. It's given that function g estimates that the seal reached its maximum depth of 302.4 meters 6 minutes after it entered the water. Therefore, function g can be expressed in vertex form as g(t)=a(t-6)2+302.4, where a is a constant. Since it's also given that the seal reached the surface of the water after 12 minutes, g(12)=0. Substituting 12 for t and 0 for g(t) in g(t)=a(t-6)2+302.4 yields 0=a(12-6)2+302.4, or 36a=-302.4. Dividing both sides of this equation by 36 gives a=-8.4. Substituting - 8.4 for a in g(t)=a(t-6)2+302.4 gives g(t)=-8.4(t-6)2+302.4. Substituting 10 for t in g(t) gives g(10)=-8.4(10-6)2+302.4, which is equivalent to g(10)=-8.4(4)2+302.4, or g(10)=168. Therefore, the estimated depth, to the nearest meter, of the seal 10 minutes after it entered the water was 168 meters.

Question 416 416 of 479 selected Nonlinear Functions H

At the time that an article was first featured on the home page of a news website, there were 40 comments on the article. An exponential model estimates that at the end of each hour after the article was first featured on the home page, the number of comments on the article had increased by 190% of the number of comments on the article at the end of the previous hour. Which of the following equations best represents this model, where C is the estimated number of comments on the article t hours after the article was first featured on the home page and t4?

  1. C=40(1.19)t

  2. C=40(1.9)t

  3. C=40(19)t

  4. C=40(2.9)t

Show Answer Correct Answer: D

Choice D is correct. It's given that an exponential model estimates that the number of comments on an article increased by a fixed percentage at the end of each hour. Therefore, the model can be represented by an exponential equation of the form C=Kat, where C is the estimated number of comments on the article t hours after the article was first featured on the home page and K and a are constants. It's also given that when the article was first featured on the home page of the news website, there were 40 comments on the article. This means that when t = 0 , C = 40 . Substituting 0 for t and 40 for C in the equation C=Kat yields 40=Ka0, or 40=K. It's also given that the number of comments on the article at the end of an hour had increased by 190% of the number of comments on the article at the end of the previous hour. Multiplying the percent increase by the number of comments on the article at the end of the previous hour yields the number of estimated additional comments the article has on its home page: (40)(190100), or 76 comments. Thus, the estimated number of comments for the following hour is the sum of the comments from the end of the previous hour and the number of additional comments, which is 40+76, or 116 . This means that when t = 1 , C = 116 . Substituting 1 for t , 116 for C , and 40 for K in the equation C=Kat yields 116=40a1, or 116=40a. Dividing both sides of this equation by 40 yields 2.9=a. Substituting 40 for K and 2.9 for a in the equation C=Kat yields C=40(2.9)t. Thus, the equation that best represents this model is C=40(2.9)t.

Choice A is incorrect. This model represents a situation where the number of comments at the end of each hour increased by 19% of the number of comments at the end of the previous hour, rather than 190%.

Choice B is incorrect. This model represents a situation where the number of comments at the end of each hour increased by 90% of the number of comments at the end of the previous hour, rather than 190%.

Choice C is incorrect. This model represents a situation where the number of comments at the end of each hour was 19 times the number of comments at the end of the previous hour, rather than increasing by 190% of the number of comments at the end of the previous hour.

Question 417 417 of 479 selected Equivalent Expressions E

Which of the following expressions is equivalent to 2 times, open parenthesis, a, b minus 3, close parenthesis, plus 2 ?

  1. 2 a, b, minus 1

  2. 2 a, b, minus 4

  3. 2 a, b, minus 5

  4. 2 a, b, minus 8

Show Answer Correct Answer: B

Choice B is correct. Applying the distributive property to the given expression yields 2 times a, b, plus, 2 times negative 3, plus 2, or 2 a, b minus 6, plus 2. Adding the like terms negative 6 and 2 results in the expression 2 a, b minus 4.

Choice A is incorrect and may result from multiplying a, b by 2 without multiplying negative 3 by 2 when applying the distributive property. Choices C and D are incorrect and may result from computational or conceptual errors.

 

Question 418 418 of 479 selected Nonlinear Functions M

  • Moving from left to right:
    • The curve is shown in quadrant 1.
    • The curve trends down sharply to x equals 3.
    • The curve then trends down gradually.
  • As x increases, the curve approaches the line y equals 0.
  • The curve passes through the following points:
    • (2 comma nine halves)
    • (3 comma 3)
    • (6 comma three halves)

The graph of the rational function f is shown, where y=f(x) and x0. Which of the following is the graph of y=f(x)+5, where x0?

    • Moving from left to right:
      • The curve passes from quadrant 1 to quadrant 4.
      • The curve trends down sharply to x equals 3. 
      • The curve then trends down gradually.
    • As x increases, the curve approaches the line y equals negative 5.
    • The curve passes through the following points:
      • (2 comma negative one half)
      • (3 comma negative 2)
      • (6 comma negative seven halves)

    • Moving from left to right:
      • The curve is shown in quadrant 1.
      • The curve trends down sharply to x equals 1. 
      • The curve then trends down gradually.
    • As x increases, the curve approaches the line y equals 0.
    • The curve passes through the following points:
      • (2 comma nine tenths)
      • (3 comma three fifths)
      • (6 comma three tenths)

    • Moving from left to right:
      • The curve is shown in quadrant 1.
      • The curve trends down sharply to x equals 2. 
      • The curve then trends down gradually.
    • As x increases, the curve approaches the line y equals 0.
    • The curve passes through the following points:
      • (2 comma 7)
      • (3 comma StartFraction 14 Over 3 EndFraction)
      • (6 comma seven thirds)

    • Moving from left to right:
      • The curve is shown in quadrant 1.
      • The curve trends down sharply to x equals 3. 
      • The curve then trends down gradually.
    • As x increases, the curve approaches the line y equals 5.
    • The curve passes through the following points:
      • (2 comma StartFraction 19 Over 2 EndFraction)
      • (3 comma 8)
      • (6 comma StartFraction 13 Over 2 EndFraction)

Show Answer Correct Answer: D

Choice D is correct. It's given that the graph of the rational function f is shown, where y=f(x) and x0. The graph shown passes through the point (3,3). It follows that when the value of x is 3 , the value of f(x) is 3 . When the value of f(x) is 3 , the value of f(x)+5 is 3+5, or 8 . Therefore, the graph of y=f(x)+5 passes through the point (3,8). Of the given choices, choice D is the only graph that passes through the point (3,8) and is therefore the graph of y=f(x)+5.

Choice A is incorrect. This is the graph of y=f(x)-5, rather than y=f(x)+5.

Choice B is incorrect. This is the graph of y=f(x)5, rather than y=f(x)+5.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 419 419 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

If 42 x = 7 x , what is the value of 7 x 2 ?

  1. 6

  2. 7

  3. 42

  4. 294

Show Answer Correct Answer: C

Choice C is correct. Multiplying both sides of the given equation by x yields 42=7x2. Therefore, the value of 7x2 is 42.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 420 420 of 479 selected Nonlinear Functions H

p(t)=90,000(1.06)t

The given function p models the population of Lowell t years after a census. Which of the following functions best models the population of Lowell m months after the census?

  1. r(m)=90,00012(1.06)m

  2. r(m)=90,000(1.0612)m

  3. r(m)=90,000(1.0612)m12

  4. r(m)=90,000(1.06)m12

Show Answer Correct Answer: D

Choice D is correct. It’s given that the function p models the population of Lowell t years after a census. Since there are 12 months in a year, m months after the census is equivalent to m12 years after the census. Substituting m12 for t in the equation p(t)=90,000(1.06)t yields p(m12)=90,000(1.06)m12. Therefore, the function r that best models the population of Lowell m months after the census is r(m)=90,000(1.06)m12.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 421 421 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

b - 72 = x y

The given equation relates the positive numbers b , x , and y . Which equation correctly expresses x in terms of b and y ?

  1. x = b - 72 y

  2. x = b y - 72

  3. x = b y - 72 y

  4. x = b y - 72 y

Show Answer Correct Answer: D

Choice D is correct. Multiplying both sides of the given equation by y yields y(b-72)=x. Distributing on the left-hand side of this equation yields by-72y=x, or x=by-72y. Therefore, the equation x=by-72y correctly expresses x in terms of b and y.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 422 422 of 479 selected Equivalent Expressions H

If k-x is a factor of the expression -x2+129nk2, where n and k are constants and k>0, what is the value of n ?

  1. -29

  2. - 1 29

  3. 1 29

  4. 29

Show Answer Correct Answer: D

Choice D is correct. If k-x is a factor of the expression -x2+(129)nk2, then the expression can be written as (k-x)(ax+b), where a and b are constants. This expression can be rewritten as akx+bk-ax2-bx, or -ax2+(ak-b)x+bk. Since this expression is equivalent to -x2+(129)nk2, it follows that - a = - 1 , a k - b = 0 , and bk=(129)nk2. Dividing each side of the equation - a = - 1 by - 1 yields a = 1 . Substituting 1 for a in the equation a k - b = 0 yields k-b=0. Adding b to each side of this equation yields k = b . Substituting k for b in the equation bk=(129)nk2 yields k2=(129)nk2. Since k is positive, dividing each side of this equation by k 2 yields 1=(129)n. Multiplying each side of this equation by 29 yields 29=n.

Alternate approach: The expression x 2 - y 2 can be written as ( x - y ) ( x + y ) , which is a difference of two squares. It follows that (129)nk2-x2 is equivalent to ((129n)k-x)((129n)k+x). It’s given that k-x is a factor of -x2+(129)nk2, so the factor (129n)k-x is equal to k-x. Adding x to both sides of the equation (129n)k-x=k-x yields (129n)k=k. Since k is positive, dividing both sides of this equation by k yields 129n=1. Squaring both sides of this equation yields 129n=1. Multiplying both sides of this equation by 29 yields n = 29 .

Choice A is incorrect. This value of n gives the expression -x2+(129)(-29)k2, or -x2-k2. This expression doesn't have k-x as a factor.

Choice B is incorrect. This value of n gives the expression -x2+(129)(-129)k2, or -x2+(-1841)k2. This expression doesn't have k-x as a factor.

Choice C is incorrect. This value of n gives the expression -x2+(129)(129)k2, or -x2+(1841)k2. This expression doesn't have k-x as a factor.

Question 423 423 of 479 selected Nonlinear Functions H

The population of a town is currently 50,000, and the population is estimated to increase each year by 3% from the previous year. Which of the following equations can be used to estimate the number of years, t, it will take for the population of the town to reach 60,000 ?

  1. 50,000 equals, 60,000 times, open parenthesis, 0 point 0 3, close parenthesis, to the power t

  2. 50,000 equals, 60,000 times, open parenthesis, 3, close parenthesis, to the power t

  3. 60,000 equals, 50,000 times, open parenthesis, 0 point 0 3, close parenthesis, to the power t

  4. 60,000 equals, 50,000 times, open parenthesis, 1 point 0 3, close parenthesis, to the power t

Show Answer Correct Answer: D

Choice D is correct. Stating that the population will increase each year by 3% from the previous year is equivalent to saying that the population each year will be 103% of the population the year before. Since the initial population is 50,000, the population after t years is given by 50,000(1.03)t. It follows that the equation 60,000 = 50,000(1.03)t can be used to estimate the number of years it will take for the population to reach 60,000.

Choice A is incorrect. This equation models how long it will take the population to decrease from 60,000 to 50,000, which is impossible given the growth factor. Choice B is incorrect and may result from misinterpreting a 3% growth as growth by a factor of 3. Additionally, this equation attempts to model how long it will take the population to decrease from 60,000 to 50,000. Choice C is incorrect and may result from misunderstanding how to model percent growth by multiplying the initial amount by a factor greater than 1.

 

Question 424 424 of 479 selected Nonlinear Functions H

For the exponential function f , the value of f(1) is k , where k is a constant. Which of the following equivalent forms of the function f shows the value of k as the coefficient or the base?

  1. f(x)=50(2)x+1

  2. f(x)=80(2)x

  3. f(x)=128(2)x-1

  4. f(x)=205(2)x-2

Show Answer Correct Answer: C

Choice C is correct. For the form of the function in choice C, f(x)=128(1.6)x-1, the value of f(1) can be found as 128(1.6)1-1, which is equivalent to 128(1.6)0, or 128 . Therefore, k = 128 , which is shown in f(x)=128(1.6)x-1 as the coefficient.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 425 425 of 479 selected Equivalent Expressions E

Which expression is equivalent to 17(x2-100y2)?

  1. 17(x-2y)(x-50y)

  2. 17(x-2y)(x+50y)

  3. 17(x-10y)(x-10y)

  4. 17(x-10y)(x+10y)

Show Answer Correct Answer: D

Choice D is correct. Expressions in the form a2-b2 follow the difference of two squares pattern and can be factored as (a-b)(a+b). In the given expression, 17(x2-100y2), the expression x2-100y2 follows the difference of two squares pattern. It follows that the expression x2-100y2 can be rewritten as (x-10y)(x+10y). Therefore, the expression 17(x-10y)(x+10y) is equivalent to 17(x2-100y2).

Choice A is incorrect. This expression is equivalent to 17(x2-52xy+100y2), not 17(x2-100y2).

Choice B is incorrect. This expression is equivalent to 17(x2+48xy-100y2), not 17(x2-100y2).

Choice C is incorrect. This expression is equivalent to 17(x2-20xy+100y2), not 17(x2-100y2).

Question 426 426 of 479 selected Nonlinear Functions M

An object is kicked from a platform. The equation h=-4.9t2+7t+9 represents this situation, where h is the height of the object above the ground, in meters, t seconds after it is kicked. Which number represents the height, in meters, from which the object was kicked?

  1. 0

  2. 4.9

  3. 7

  4. 9

Show Answer Correct Answer: D

Choice D is correct. It’s given that the equation h = - 4.9 t 2 + 7 t + 9 represents this situation, where h is the height, in meters, of the object t seconds after it is kicked. It follows that the height, in meters, from which the object was kicked is the value of h when t = 0 . Substituting 0 for t in the equation h = - 4.9 t 2 + 7 t + 9 yields h=-4.9(0)2+7(0)+9, or h = 9 . Therefore, the object was kicked from a height of 9 meters.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 427 427 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

y - 57 = p x

The given equation relates the positive numbers p , x , and y . Which equation correctly expresses y in terms of p and x ?

  1. y=57x+p

  2. y = p x + 57

  3. y = 57 p x

  4. y = p x 57

Show Answer Correct Answer: B

Choice B is correct. Adding 57 to each side of the given equation yields y = p x + 57 . Therefore, the equation y = p x + 57 correctly expresses y in terms of p and x .

Choice A is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 428 428 of 479 selected Equivalent Expressions M

(5x3-3)-(-4x3+8)

The given expression is equivalent to b x 3 - 11 , where b is a constant. What is the value of b ?

Show Answer Correct Answer: 9

The correct answer is 9 . The given expression can be rewritten as (5x3-3)+(-1)(-4x3+8). By applying the distributive property, this expression can be rewritten as 5x3-3+4x3+(-8), which is equivalent to (5x3+4x3)+(-3+(-8)). Adding like terms in this expression yields 9x3-11. Since it's given that (5x3-3)-(-4x3+8) is equivalent to bx3-11, it follows that 9x3-11 is equivalent to bx3-11. Therefore, the coefficients of x3 in these two expressions must be equivalent, and the value of b must be 9 .

Question 429 429 of 479 selected Equivalent Expressions E

Which expression is equivalent to 9x+6x+2y+3y?

  1. 3 x + 5 y

  2. 6 x + 8 y

  3. 12 x + 8 y

  4. 15 x + 5 y

Show Answer Correct Answer: D

Choice D is correct. Combining like terms in the given expression yields (9x+6x)+(2y+3y), or 15x+5y.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 430 430 of 479 selected Nonlinear Functions H

h of x equals, negative 16 x squared, plus 100 x, plus 10

The quadratic function above models the height above the ground h, in feet, of a projectile x seconds after it had been launched vertically. If y equals h of x is graphed in the xy-plane, which of the following represents the real-life meaning of the positive x-intercept of the graph?

  1. The initial height of the projectile

  2. The maximum height of the projectile

  3. The time at which the projectile reaches its maximum height

  4. The time at which the projectile hits the ground

Show Answer Correct Answer: D

Choice D is correct. The positive x-intercept of the graph of y equals, h of x is a point with coordinates x comma y for which y equals 0. Since y equals, h of x models the height above the ground, in feet, of the projectile, a y-value of 0 must correspond to the height of the projectile when it is 0 feet above ground or, in other words, when the projectile is on the ground. Since x represents the time since the projectile was launched, it follows that the positive x-intercept, the point with coordinates x comma 0, represents the time at which the projectile hits the ground.

Choice A is incorrect and may result from misidentifying the y-intercept as a positive x-intercept. Choice B is incorrect and may result from misidentifying the y-value of the vertex of the graph of the function as an x-intercept. Choice C is incorrect and may result from misidentifying the x-value of the vertex of the graph of the function as an x-intercept.

 

Question 431 431 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

  • For the absolute value function in the system:
    • Moving from left to right:
      • The function slants sharply down to the point (negative 3 comma 4).
      • The function then slants sharply up.
    • The function passes through the following points:
      • (negative seven halves comma nine halves)
      • (negative 3 comma 4)
      • (negative five halves comma nine halves)
  • For the linear function in the system:
    • The function slants sharply up from left to right.
    • The function passes through the following points:
      • (negative seven halves comma nine halves)
      • (0 comma 8)

The graph of a system of an absolute value function and a linear function is shown. What is the solution (x,y) to this system of two equations?

  1. (0,8)

  2. (72,92)

  3. (-72,92)

  4. (-3,4)

Show Answer Correct Answer: C

Choice C is correct. The solution to a system of two equations corresponds to the point where the graphs of the equations intersect. The graphs of the linear function and the absolute value function shown intersect at a point with an x-coordinate between - 4 and - 3 and a y-coordinate between 4 and 5 . Of the given choices, only (-72,92) has an x-coordinate between - 4 and - 3 and a y-coordinate between 4 and 5 .

Choice A is incorrect. This is the y-intercept of the graph of the linear function.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the vertex of the graph of the absolute value function.

Question 432 432 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

y = 4 x

y = x 2 - 12

A solution to the given system of equations is (x,y), where x>0. What is the value of x ?

Show Answer Correct Answer: 6

The correct answer is 6 . It’s given that y = 4 x and y = x 2 - 12 . Since y = 4 x , substituting 4 x for y in the second equation of the given system yields 4 x = x 2 - 12 . Subtracting 4 x from both sides of this equation yields 0 = x 2 - 4 x - 12 . This equation can be rewritten as 0=(x-6)(x+2). By the zero product property, x - 6 = 0 or x + 2 = 0 . Adding 6 to both sides of the equation x - 6 = 0 yields x = 6 . Subtracting 2 from both sides of the equation x + 2 = 0 yields x = -2 . Therefore, solutions to the given system of equations occur when x = 6 and when x = -2 . It’s given that a solution to the given system of equations is (x,y), where x>0. Since 6 is greater than 0 , it follows that the value of x is 6 .

Question 433 433 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

y equals, x squared, minus 1 and y equals 3

When the equations above are graphed in the xy-plane, what are the coordinates (x, y) of the points of intersection of the two graphs?

  1. 2 comma 3

    and negative 2 comma 3

  2. 2 comma 4

    and negative 2 comma 4

  3. 3 comma 8

    and negative 3 comma 8

  4. the square root of 2 comma 3

    and the negative square root of 2 comma 3

Show Answer Correct Answer: A

Choice A is correct. The two equations form a system of equations, and the solutions to the system correspond to the points of intersection of the graphs. The solutions to the system can be found by substitution. Since the second equation gives y = 3, substituting 3 for y in the first equation gives 3 = x2 – 1. Adding 1 to both sides of the equation gives 4 = x2. Solving by taking the square root of both sides of the equation gives x = ±2. Since y = 3 for all values of x for the second equation, the solutions are (2, 3) and (–2, 3). Therefore, the points of intersection of the two graphs are (2, 3) and (–2, 3).

Choices B, C, and D are incorrect and may be the result of calculation errors.

Question 434 434 of 479 selected Nonlinear Functions E

The function f is defined by f(x)=5x2. What is the value of f(8)?

  1. 40

  2. 50

  3. 80

  4. 320

Show Answer Correct Answer: D

Choice D is correct. It's given that the function f is defined by f(x)=5x2. Substituting 8 for x in f(x)=5x2 yields f(8)=5(8)2, which is equivalent to f(8)=5(64), or f(8)=320. Therefore, the value of f(8) is 320 .

Choice A is incorrect. This is the value of f(8) if f(x)=5x.

Choice B is incorrect. This is the value of f(8) if f(x)=5(x+2).

Choice C is incorrect. This is the value of f(8) if f(x)=(5x)(2).

Question 435 435 of 479 selected Equivalent Expressions E

Which expression is equivalent to 34 x + 34 y ?

  1. 34 x y

  2. 34(x+y)

  3. 68 y

  4. 68 x

Show Answer Correct Answer: B

Choice B is correct. Since 34 is a common factor of each term in the given expression, the expression can be rewritten as 34(x+y).

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This expression is equivalent to 34y+34y.

Choice D is incorrect. This expression is equivalent to 34x+34x.

Question 436 436 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

If |4x-4|=112, what is the positive value of x - 1 ?

Show Answer Correct Answer: 28

The correct answer is 28 . The given absolute value equation can be rewritten as two linear equations: 4 x - 4 = 112 and -(4x-4)=112, or 4 x - 4 = -112 . Adding 4 to both sides of the equation 4 x - 4 = 112 results in 4 x = 116 . Dividing both sides of this equation by 4 results in x = 29 . Adding 4 to both sides of the equation 4 x - 4 = -112 results in 4 x = -108 . Dividing both sides of this equation by 4 results in x = -27 . Therefore, the two values of x - 1 are 29-1, or 28 , and -27-1, or -28 . Thus, the positive value of x - 1 is 28 .

Alternate approach: The given equation can be rewritten as |4(x-1)|=112, which is equivalent to 4|x-1|=112. Dividing both sides of this equation by 4 yields |x-1|=28. This equation can be rewritten as two linear equations: x - 1 = 28  and -(x-1)=28, or x - 1 = -28 . Therefore, the positive value of x - 1 is 28 .

Question 437 437 of 479 selected Nonlinear Functions H
xf of x
1a
2a, to the fifth power.
3a, to the ninth power


For the exponential function f, the table above shows several values of x and their corresponding values of f of x, where a is a constant greater than 1. If k is a constant and f of k, equals a, to the twenty ninth power, what is the value of k ?

Show Answer

The correct answer is 8. The values of f of x for the exponential function f shown in the table increase by a factor of a, to the fourth power for each increase of 1 in x. This relationship can be represented by the equation f of x equals, a, raised to the 4 x plus b power, where b is a constant. It’s given that when x equals 2, f of x equals, a, to the fifth power.  Substituting 2 for x and a, to the fifth power for f of x into f of x equals, a, raised to the 4 x plus b power  yields a, to the fifth power equals, a, raised to the 4 times 2, plus b power. Since 4 times 2, plus b, equals 5, it follows that b equals negative 3. Thus, an equation that defines the function f is f of x equals, a, raised to the 4 x minus 3 power. It follows that the value of k such that f of k equals, a, to the twenty ninth power can be found by solving the equation 4 k minus 3, equals 29, which yields k equals 8.

Question 438 438 of 479 selected Equivalent Expressions E

Which expression is equivalent to 256 w 2 - 676 ?

  1. (16w-26)(16w-26)

  2. ( 8 w - 13 ) ( 8 w + 13 )

  3. (8w-13)(8w-13)

  4. ( 16 w - 26 ) ( 16 w + 26 )

Show Answer Correct Answer: D

Choice D is correct. The given expression follows the difference of two squares pattern, x2-y2, which factors as (x-y)(x+y). Therefore, the expression 256w2-676 can be written as (16w)2-262, or (16w)(16w)-(26)(26), which factors as (16w-26)(16w+26).

Choice A is incorrect. This expression is equivalent to 256w2-832w+676.

Choice B is incorrect. This expression is equivalent to 64w2-169.

Choice C is incorrect. This expression is equivalent to 64w2-208w+169.

Question 439 439 of 479 selected Nonlinear Functions E

h(x)=x2-3

Which table gives three values of x and their corresponding values of h(x) for the given function h ?

Show Answer Correct Answer: B

Choice B is correct. It′s given that h(x)=x2-3. Each table gives 1 , 2 , and 3 as the three given values of x . Substituting 1 for x in the equation h(x)=x2-3 yields h(1)=(1)2-3, or h(1)=-2. Substituting 2 for x in the equation h(x)=x2-3 yields h(2)=(2)2-3, or h(2)=1. Finally, substituting 3 for x in the equation h(x)=x2-3 yields h(3)=(3)2-3, or h(3)=6. Therefore, h(x) is -2 when x is 1 , h(x) is 1 when x is 2 , and h(x) is 6 when x is 3 . Choice B is a table with these values of x and their corresponding values of h(x).

Choice A is incorrect. This is a table of values for the function h(x)=x+3, not h(x)=x2-3.

Choice C is incorrect. This is a table of values for the function h(x)=2x-3, not h(x)=x2-3.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 440 440 of 479 selected Nonlinear Functions M

Bacteria are growing in a liquid growth medium. There were 300,000 cells per milliliter during an initial observation. The number of cells per milliliter doubles every 3 hours. How many cells per milliliter will there be 15 hours after the initial observation?

  1. 1,500,000

  2. 2,400,000

  3. 4,500,000

  4. 9,600,000

Show Answer Correct Answer: D

Choice D is correct. Let y represent the number of cells per milliliter x hours after the initial observation. Since the number of cells per milliliter doubles every 3 hours, the relationship between x and y can be represented by an exponential equation of the form y=a(b)xk, where a is the number of cells per milliliter during the initial observation and the number of cells per milliliter increases by a factor of b every k hours. It’s given that there were 300,000 cells per milliliter during the initial observation. Therefore, a = 300,000 . It’s also given that the number of cells per milliliter doubles, or increases by a factor of 2 , every 3 hours. Therefore, b = 2 and k = 3 . Substituting 300,000 for a , 2 for b , and 3 for k in the equation y=a(b)xk yields y=300,000(2)x3. The number of cells per milliliter there will be 15 hours after the initial observation is the value of y in this equation when x = 15 . Substituting 15 for x in the equation y=300,000(2)x3 yields y=300,000(2)153, or y=300,000(2)5. This is equivalent to y=300,000(32), or y = 9,600,000 . Therefore, 15 hours after the initial observation, there will be 9,600,000 cells per milliliter.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 441 441 of 479 selected Equivalent Expressions E

5 x plus 15

Which of the following is equivalent to the given expression?

  1. 5 times, open parenthesis, x plus 3, close parenthesis

  2. 5 times, open parenthesis, x plus 10, close parenthesis

  3. 5 times, open parenthesis, x plus 15, close parenthesis

  4. 5 times, open parenthesis, x plus 20, close parenthesis

Show Answer Correct Answer: A

Choice A is correct. Since 5 is a factor of both terms, 5 x and 15, the given expression can be factored and rewritten as 5 times, open parenthesis, x plus 3, close parenthesis.

Choice B is incorrect and may result from subtracting 5 from the constant when factoring 5 from the given expression. Choice C is incorrect and may result from factoring 5 from only the first term, not both terms, of the given expression. Choice D is incorrect and may result from adding 5 to the constant when factoring 5 from the given expression.

 

Question 442 442 of 479 selected Nonlinear Functions H

  • Moving from left to right:
    • The curve is in quadrant 3.
    • The curve trends down sharply.
  • As x increases, the curve approaches the line x equals negative 4.
  • As x decreases, the curve approaches the line y equals 0.
  • The curve passes through the following points:
    • (negative 7 comma negative 2)
    • (negative 6 comma negative 3)
    • (negative 5 comma negative 6)

The rational function f is defined by an equation in the form f(x)=ax+b, where a and b are constants. The partial graph of y=f(x) is shown. If  g(x)=f(x+4), which equation could define function g ?

  1. g(x)=6x

  2. g(x)=6x+4

  3. g(x)=6x+8

  4. g(x)=6(x+4)x+4

Show Answer Correct Answer: C

Choice C is correct. It's given that f(x)=ax+b and that the graph shown is a partial graph of y=f(x). Substituting y for f(x) in the equation f(x)=ax+b yields y = a x + b . The graph passes through the point (-7,-2). Substituting -7 for x and -2 for y in the equation y = a x + b yields -2=a-7+b. Multiplying each side of this equation by -7+b yields -2(-7+b)=a, or 14-2b=a. The graph also passes through the point (-5,-6). Substituting -5 for x and -6 for y in the equation y = a x + b yields -6=a-5+b. Multiplying each side of this equation by -5+b yields -6(-5+b)=a, or 30-6b=a. Substituting 14-2b for a in this equation yields 30-6b=14-2b. Adding 6 b to each side of this equation yields 30=14+4b. Subtracting 14 from each side of this equation yields 16 = 4 b . Dividing each side of this equation by 4 yields 4 = b . Substituting 4 for b in the equation 14-2b=a yields 14-2(4)=a, or 6 = a . Substituting 6 for a and 4 for b in the equation f(x)=ax+b yields f(x)=6x+4. It's given that g(x)=f(x+4). Substituting x + 4 for x in the equation f(x)=6x+4 yields f(x+4)=6x+4+4, which is equivalent to f(x+4)=6x+8. It follows that g(x)=6x+8.

Choice A is incorrect. This could define function g if g(x)=f(x-4).

Choice B is incorrect. This could define function g if g(x)=f(x).

Choice D is incorrect. This could define function g if g(x)=f(x)·(x+4).

Question 443 443 of 479 selected Nonlinear Functions E

The function f is defined by f(x)=16x. What is the value of f(x) when x = 17 ?

  1. 1617

  2. 1716

  3. 16

  4. 17

Show Answer Correct Answer: A

Choice A is correct. It's given that f(x)=16x. Substituting 17 for x in this function yields f(17)=1617. Therefore, when x = 17 , the value of f(x) is 1617.

Choice B is incorrect. This is the value of the reciprocal of f(x) when x = 17 .

Choice C is incorrect. This is the value of f(x) when x = 1 .

Choice D is incorrect. This is the value of x when x = 17 .

Question 444 444 of 479 selected Nonlinear Functions H

The area of a triangle is equal to x 2 square centimeters. The length of the base of the triangle is 2 x + 22 centimeters, and the height of the triangle is x - 10 centimeters. What is the value of x ?

Show Answer Correct Answer: 110

The correct answer is 110. The area of a triangle is equal to one half of the product of the length of the base of the triangle and the height of the triangle. It's given that the length of the base of the triangle is 2x+22 centimeters and the height of the triangle is x-10 centimeters; therefore, its area is 12(2x+22)(x-10) square centimeters. It's also given that the area of the triangle is equal to x2 square centimeters. Therefore, it follows that 12(2x+22)(x-10)=x2.  This equation can be rewritten as (x+11)(x-10)=x2, or x2+x-110=x2. Subtracting x2 from both sides of this equation yields x-110=0. Adding 110 to both sides of this equation yields x=110. Therefore, the value of x is 110.

Question 445 445 of 479 selected Nonlinear Functions H

f(x)= ( x - 10 ) ( x + 13 )

The function f is defined by the given equation. For what value of x does f(x) reach its minimum?

  1. -130

  2. -13

  3. - 23 2

  4. - 3 2

Show Answer Correct Answer: D

Choice D is correct. It's given that f(x)=(x-10)(x+13), which can be rewritten as  f(x)=x2+3x-130. Since the coefficient of the x2-term is positive, the graph of y=f(x) in the xy-plane opens upward and reaches its minimum value at its vertex. The x-coordinate of the vertex is the value of x such that f(x) reaches its minimum. For an equation in the form f(x)=ax2+bx+c, where a , b , and c are constants, the x-coordinate of the vertex is -b2a. For the equation f(x)=x2+3x-130, a=1, b=3, and c=-130. It follows that the x-coordinate of the vertex is -32(1), or -32. Therefore, f(x) reaches its minimum when the value of x is -32.

Alternate approach: The value of x for the vertex of a parabola is the x-value of the midpoint between the two x-intercepts of the parabola. Since it’s given that f(x)=(x-10)(x+13), it follows that the two x-intercepts of the graph of y=f(x) in the xy-plane occur when x=10 and x=-13, or at the points (10,0) and (-13,0). The midpoint between two points, (x1,y1) and (x2,y2), is (x1+x22,y1+y22). Therefore, the midpoint between (10,0) and (-13,0) is (10+(-13)2,0+02), or (-32,0). It follows that f(x) reaches its minimum when the value of x is -32.

Choice A is incorrect. This is the y-coordinate of the y-intercept of the graph of y=f(x) in the xy-plane.

Choice B is incorrect. This is one of the x-coordinates of the x-intercepts of the graph of y=f(x) in the xy-plane.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 446 446 of 479 selected Equivalent Expressions M

Which expression is equivalent to x 9 y 9 7 , where x and y are positive?

  1. (xy)79

  2. (xy)97

  3. (xy)16

  4. (xy)63

Show Answer Correct Answer: B

Choice B is correct. For positive values of a and b , ambm=(ab)m, an=(a)1n, and (aj)k=ajk. Therefore, the given expression, x9y97, can be rewritten as (xy)97. This expression is equivalent to ((xy)9)17, which can be rewritten as (xy)9·17, or (xy)97

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 447 447 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

12n-2t=-2w

The given equation relates the variables n , t , and w , where n>0, t>0, and w>t. Which expression is equivalent to n ?

  1. 12tw

  2. 6(t-w)

  3. w-t6tw

  4. 6tww-t

Show Answer Correct Answer: D

Choice D is correct. Adding 2 t to each side of the given equation yields  12 n = - 2 w + 2 t . The fractions on the right side of this equation have a common denominator of t w ; therefore, the equation can be written as 12n=2wtw-2ttw, or 12n=2w-2ttw, which is equivalent to 12n=2(w-t)tw. Dividing each side of this equation by 2 yields 6n=w-ttw. Since n , t , w , and w-t are all positive quantities, taking the reciprocal of each side of the equation 6n=w-ttw yields an equivalent equation: n6=tww-t. Multiplying each side of this equation by 6 yields n=6tww-t.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is equivalent to 1 n rather than n .

Question 448 448 of 479 selected Nonlinear Functions M

m(t)=-0.0274(t7)2+7.3873(t7)+75.032

The function m gives the predicted body mass m(t), in kilograms (kg), of a certain animal t days after it was born in a wildlife reserve, where t390. Which of the following is the best interpretation of the statement “m(330) is approximately equal to 362 ” in this context?

  1. The predicted body mass of the animal was approximately 330 kg 362 days after it was born.

  2. The predicted body mass of the animal was approximately 362 kg 330 days after it was born.

  3. The predicted body mass of the animal was approximately 362 kg 3307 days after it was born.

  4. The predicted body mass of the animal was approximately 3307 kg 362 days after it was born.

Show Answer Correct Answer: B

Choice B is correct. In the statement "m(330) is approximately equal to 362 ," the input of the function, 330 , is the value of t , the elapsed time, in days, since the animal was born. The approximate value of the function, 362 , is the predicted body mass, in kilograms, of the animal after that time has elapsed. Therefore, the predicted body mass of the animal was approximately 362 kg 330 days after it was born.

Choice A is incorrect. This would be the best interpretation of the statement "m(362) is approximately equal to 330 ."

Choice C is incorrect. The number 3307 is the number of weeks, not the number of days, after the animal was born.

Choice D is incorrect. This would be the best interpretation of the statement "m(362) is approximately equal to 3307."

Question 449 449 of 479 selected Nonlinear Functions H

y=2(x-d )(x+d )(x+g )(x-d )

In the given equation, d and g are unique positive constants. When the equation is graphed in the xy-plane, how many distinct x-intercepts does the graph have?

  1. 4

  2. 3

  3. 2

  4. 1

Show Answer Correct Answer: B

Choice B is correct. An x-intercept of a graph in the xy-plane is a point on the graph where the value of y is 0 . Substituting 0 for y in the given equation yields 0=2(x-d)(x+d)(x+g)(x-d). By the zero product property, the solutions to this equation are x = d , x = - d , x = - g , and x = d . However, x = d and x = d are identical. It's given that d and g are unique positive constants. It follows that the equation 0=2(x-d)(x+d)(x+g)(x-d) has 3 unique solutions: x = d , x = - d , and x = - g . Thus, the graph of the given equation has 3 distinct x-intercepts.

Choice A is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 450 450 of 479 selected Nonlinear Functions H

The functions g and h are defined by the given equations, where x0. Which of the following equations displays, as a constant or coefficient, the minimum value of the function it defines, where x0?

  1. g(x)=18(1.16)(1.4)x+2
  2. h(x)=18(1.4)x+4
  1. I only

  2. II only

  3. I and II

  4. Neither I nor II

Show Answer Correct Answer: D

Choice D is correct. A function defined by an equation in the form f(x)=a(b)x+h, where a, b, and h are positive constants and x0, has a minimum value of f(0). It's given that function g is defined by g(x)=18(1.16)(1.4)x+2, which is equivalent to g(x)=20.88(1.4)x+2. Substituting 0 for x in this equation yields g(0)=20.88(1.4)0+2, or g(0)=40.9248. Therefore, the minimum value of g(x) is 40.9248, so g(x)=18(1.16)(1.4)x+2 doesn't display its minimum value as a constant or coefficient. It's also given that function h is defined by h(x)=18(1.4)x+4. Substituting 0 for x in this equation yields h(0)=18(1.4)0+4, or h(0)=69.1488. Therefore, the minimum value of h(x) is 69.1488, so h(x)=18(1.4)x+4 doesn't display its minimum value as a constant or coefficient. Therefore, neither I nor II displays, as a constant or coefficient, the minimum value of the function it defines, where x0.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 451 451 of 479 selected Nonlinear Functions M

The area A , in square centimeters, of a rectangular cutting board can be represented by the expression w(w+9), where w is the width, in centimeters, of the cutting board. Which expression represents the length, in centimeters, of the cutting board?

  1. w(w+9)

  2. w

  3. 9

  4. (w+9)

Show Answer Correct Answer: D

Choice D is correct. It's given that the expression w(w+9) represents the area, in square centimeters, of a rectangular cutting board, where w is the width, in centimeters, of the cutting board. The area of a rectangle can be calculated by multiplying its length by its width. It follows that the length, in centimeters, of the cutting board is represented by the expression (w+9)

Choice A is incorrect. This expression represents the area, in square centimeters, of the cutting board, not its length, in centimeters.

Choice B is incorrect. This expression represents the width, in centimeters, of the cutting board, not its length.

Choice C is incorrect. This is the difference between the length, in centimeters, and the width, in centimeters, of the cutting board, not its length, in centimeters.

Question 452 452 of 479 selected Equivalent Expressions E

Which expression is equivalent to 16(x+15)?

  1. 16 x + 31

  2. 16 x + 240

  3. 16 x + 1

  4. 16 x + 15

Show Answer Correct Answer: B

Choice B is correct. The expression 16(x+15) can be rewritten as 16(x)+16(15), which is equivalent to 16x+240.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 453 453 of 479 selected Nonlinear Functions H

For the function q , the value of q(x) decreases by 45 % for every increase in the value of x by 1 . If q(0)=14, which equation defines q ?

  1. q(x)=0.55(14)x

  2. q(x)=1.45(14)x

  3. q(x)=14(0.55)x

  4. q(x)=14(1.45)x

Show Answer Correct Answer: C

Choice C is correct. Since the value of q(x) decreases by a fixed percentage, 45%, for every increase in the value of x by 1 , the function q is a decreasing exponential function. A decreasing exponential function can be written in the form q(x)=a(1-p100)x, where a is the value of q(0) and the value of q(x) decreases by p% for every increase in the value of x by 1 . If q(0)=14, then a = 14 . Since the value of q(x) decreases by 45% for every increase in the value of x by 1 , p = 45 . Substituting 14 for a and 45 for p in the equation q(x)=a(1-p100)x yields q(x)=14(1-45100)x, which is equivalent to q(x)=14(1-0.45)x, or q(x)=14(0.55)x.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. For this function, the value of q(x) increases, rather than decreases, by 45% for every increase in the value of x by 1 .

Question 454 454 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

In the xy-plane, the graph of the equation y = - x 2 + 9 x - 100 intersects the line y = c at exactly one point. What is the value of c ?

  1. - 481 4

  2. -100

  3. - 319 4

  4. - 9 2

Show Answer Correct Answer: C

Choice C is correct. In the xy-plane, the graph of the line y=c is a horizontal line that crosses the y-axis at y=c and the graph of the quadratic equation y=-x2+9x-100 is a parabola. A parabola can intersect a horizontal line at exactly one point only at its vertex. Therefore, the value of c should be equal to the y-coordinate of the vertex of the graph of the given equation. For a quadratic equation in vertex form, y=a(x-h)2+k, the vertex of its graph in the xy-plane is (h,k). The given quadratic equation, y=-x2+9x-100, can be rewritten as y=-(x2-2(92)x+(92)2)+(92)2-100, or y=-(x-92)2+(-3194). Thus, the value of c is equal to -3194.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

 

Question 455 455 of 479 selected Equivalent Expressions M

open parenthesis, x minus 11 y, close parenthesis, times, open parenthesis, 2 x minus 3 y, close parenthesis, minus, 12 y, times, open parenthesis, negative 2 x, plus 3 y, close parenthesis

Which of the following is equivalent to the expression above?

  1. x minus 23 y

  2. 2 x squared, minus x y, minus 3 y squared

  3. 2 x squared, plus 24 x y, plus 36 y squared

  4. 2 x squared, minus 49 x y, plus 69 y squared

Show Answer Correct Answer: B

Choice B is correct. Expanding all terms yields (x – 11y)(2x – 3y) – 12y(–2x + 3y), which is equivalent to 2x2 – 22xy – 3xy + 33y2 + 24xy – 36y2. Combining like terms gives 2x2xy – 3y2.

Choice A is incorrect and may be the result of using the sums of the coefficients of the existing x and y terms as the coefficients of the x and y terms in the new expressions. Choice C is incorrect and may be the result of incorrectly combining like terms. Choice D is incorrect and may be the result of using the incorrect sign in front of the 12y term.

 

Question 456 456 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

b = 42 c f

The given equation relates the positive numbers b , c , and f . Which equation correctly expresses c in terms of b and f ?

  1. c = b 42 f

  2. c = b - 42 f

  3. c = 42 b f

  4. c=42-b-f

Show Answer Correct Answer: A

Choice A is correct. It's given that the equation b=42cf relates the positive numbers b , c , and f . Dividing each side of the given equation by 42 f yields b42f=c, or c=b42f. Thus, the equation c=b42f correctly expresses c in terms of b and f .

Choice B is incorrect. This equation can be rewritten as b=cf+42.

Choice C is incorrect. This equation can be rewritten as b=c42f.

Choice D is incorrect. This equation can be rewritten as b=42-c-f.

Question 457 457 of 479 selected Nonlinear Functions H

Two variables, x and y , are related such that for each increase of 1 in the value of x , the value of y increases by a factor of 4 . When x = 0 , y = 200 . Which equation represents this relationship?

  1. y=4(x)200

  2. y=4(200)x

  3. y=200(x)4

  4. y=200(4)x

Show Answer Correct Answer: D

Choice D is correct. Since the value of y increases by a constant factor, 4 , for each increase of 1 in the value of x , the relationship between x and y is exponential. An exponential relationship between x and y can be represented by an equation of the form y=a(b)x, where a is the value of x when y = 0 and y increases by a factor of b for each increase of 1 in the value of x . Since y = 200 when x = 0 , a = 200 . Since y increases by a factor of 4 for each increase of 1 in the value of x , b = 4 . Substituting 200 for a and 4 for b in the equation y=a(b)x yields y=200(4)x. Thus, the equation y=200(4)x represents the relationship between x and y .

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect. This equation represents a relationship where for each increase of 1 in the value of x , the value of y increases by a factor of 200 , not 4 , and when x = 0 , y is equal to 4 , not 200 .

Choice C is incorrect and may result from conceptual errors.

Question 458 458 of 479 selected Nonlinear Functions E

  • The parabola opens upward.
  • The parabola passes through the following points:
    • (negative 1 comma 5)
    • (0 comma 2)
    • (1 comma 5)

The graph of the quadratic function y=f(x) is shown. What is the vertex of the graph?

  1. (0,-2)

  2. (0,-3)

  3. (0,2)

  4. (0,3)

Show Answer Correct Answer: C

Choice C is correct. The vertex of the graph of a quadratic function in the xy-plane is the point at which the graph is either at its minimum or maximum y-value. In the graph shown, the minimum y-value occurs at the point (0,2).

Choice A is incorrect. The graph shown doesn't pass through the point (0,-2).

Choice B is incorrect. The graph shown doesn't pass through the point (0,-3).

Choice D is incorrect. The graph shown doesn't pass through the point (0,3).

Question 459 459 of 479 selected Equivalent Expressions E

Which expression is equivalent to 12x3-5x3?

  1. 7 x 6

  2. 17 x 3

  3. 7 x 3

  4. 17 x 6

Show Answer Correct Answer: C

Choice C is correct. The given expression shows subtraction of two like terms. The two terms can be subtracted as follows: 12x3-5x3=(12-5)x3, or 7x3.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the result of adding, not subtracting, the two like terms.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 460 460 of 479 selected Nonlinear Functions M

f(x)=a-19x+5

In the given function f , a is a constant. The graph of function f in the xy-plane, where y=f(x), is translated 3 units down and 4 units to the right to produce the graph of y=g(x). Which equation defines function g ?

  1. g(x)=a-19x+4+2

  2. g(x)=a-19x-4+2

  3. g(x)=a-22x+4+5

  4. g(x)=a-22x-4+5

Show Answer Correct Answer: B

Choice B is correct. It's given that the graph of y=g(x) is produced by translating the graph of y=f(x) 3 units down and 4 units to the right in the xy-plane. Therefore, function g can be defined by an equation in the form g(x)=f(x-4)-3. Function f is defined by the equation f(x)=a-19x+5, where a is a constant. Substituting x-4 for x in the equation f(x)=a-19x+5 yields f(x-4)=a-19x-4+5. Substituting a-19x-4+5 for f(x-4) in the equation g(x)=f(x-4)-3 yields g(x)=a-19x-4+5-3, or g(x)=a-19x-4+2. Therefore, the equation that defines function g is g(x)=a-19x-4+2.

Choice A is incorrect. This equation defines a function whose graph is produced by translating the graph of y=f(x) 3 units down and 4 units to the left, not 3 units down and 4 units to the right.

Choice C is incorrect. This equation defines a function whose graph is produced by translating the graph of  y=f(x) 4 units to the left, not 3 units down and 4 units to the right.

Choice D is incorrect. This equation defines a function whose graph is produced by translating the graph of y=f(x) 4 units to the right, not 3 units down and 4 units to the right.

Question 461 461 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

c-7=25p+k

The given equation relates the positive numbers c , p , and k . Which equation correctly expresses c in terms of p and k ?

  1. c=25p+k+7

  2. c=25p+k-7

  3. c=7(25p+k)

  4. c=25p+k7

Show Answer Correct Answer: A

Choice A is correct. Adding 7 to each side of the given equation yields c=25p+k+7.

Choice B is incorrect. This equation is equivalent to c+7=25p+k, not c-7=25p+k.

Choice C is incorrect. This equation is equivalent to c7=25p+k, not c-7=25p+k.

Choice D is incorrect. This equation is equivalent to 7c=25p+k, not c-7=25p+k.

Question 462 462 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

If open parenthesis, x plus 5, close parenthesis, squared, equals 4, which of the following is a possible value of x ?

  1. 1

  2. negative 1

  3. negative 2

  4. negative 3

Show Answer Correct Answer: D

Choice D is correct. If open parenthesis, x plus 5, close parenthesis, squared, equals 4, then taking the square root of each side of the equation gives x plus 5, equals 2 or x plus 5, equals negative 2. Solving these equations for x gives x equals negative 3 or x equals negative 7. Of these, negative 3 is the only solution given as a choice.

Choice A is incorrect and may result from solving the equation x plus 5, equals 4 and making a sign error. Choice B is incorrect and may result from solving the equation x plus 5, equals 4. Choice C is incorrect and may result from finding a possible value of x plus 5.

 

Question 463 463 of 479 selected Nonlinear Functions H

f(x)=(1.84)x4

The function f is defined by the given equation. The equation can be rewritten as f(x)=(1+p100)x, where p is a constant. Which of the following is closest to the value of p ?

  1. 16

  2. 21

  3. 46

  4. 96

Show Answer Correct Answer: A

Choice A is correct. The equation f(x)=(1.84)x4 can be rewritten as f(x)=(1.84)(14)(x), which is equivalent to f(x)=(1.8414)x, or approximately f(x)=(1.16467)x. Since it's given that f(x)=(1.84)x4 can be rewritten as f(x)=(1+p100)x, where p is a constant, it follows that 1+p100 is approximately equal to 1.16467. Therefore, p100 is approximately equal to 0.16467. It follows that the value of p is approximately equal to 16.467. Of the given choices, 16 is closest to the value of p .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 464 464 of 479 selected Nonlinear Functions H

What is the minimum value of the function f defined by f of x equals, open parenthesis, x minus 2, close parenthesis, squared, minus 4  ?

  1. negative 4

  2. negative 2

  3. 2

  4. 4

Show Answer Correct Answer: A

Choice A is correct. The given quadratic function f is in vertex form, f of x equals, open parenthesis, x minus h, close parenthesis, squared, plus k , where the point with coordinates h comma k is the vertex of the graph of y equals f of x in the xy-plane. Therefore, the vertex of the graph of y equals f of x is the point with coordinates 2 comma negative 4. In addition, the y-coordinate of the vertex represents either the minimum or maximum value of a quadratic function, depending on whether the graph of the function opens upward or downward. Since the leading coefficient of f (the coefficient of the term open parenthesis, x minus 2, close parenthesis, squared) is 1, which is positive, the graph of y equals f of x opens upward. It follows that at x equals 2 , the minimum value of the function f is negative 4 .

Choice B is incorrect and may result from making a sign error and from using the x-coordinate of the vertex. Choice C is incorrect and may result from using the x-coordinate of the vertex. Choice D is incorrect and may result from making a sign error.

 

Question 465 465 of 479 selected Nonlinear Functions M

A function p estimates that there were 2,000 animals in a population in 1998. Each year from 1998 to 2010, the function estimates that the number of animals in this population increased by 3% of the number of animals in the population the previous year. Which equation defines this function, where p(x) is the estimated number of animals in the population x years after 1998?

  1. p(x)=2,000(3)x

  2. p(x)=2,000(1.97)x

  3. p(x)=2,000(1.03)x

  4. p(x)=2,000(0.97)x

Show Answer Correct Answer: C

Choice C is correct. It's given that a function p estimates that there were 2,000 animals in a population in 1998 and that each year from 1998 to 2010, the number of animals in this population increased by 3% of the number of animals in the population the previous year. It follows that this situation can be represented by the function p(x)=a(1+r100)x, where p(x) is the estimated number of animals in the population x years after 1998, a is the estimated number of animals in the population in 1998, and each year the estimated number of animals increased by r%. Substituting 2,000 for a and 3 for r in this function yields p(x)=2,000(1+3100)x, or p(x)=2,000(1.03)x.

Choice A is incorrect. This function represents a population in which each year the number of animals increased by 200%, not 3%, of the number of animals in the population the previous year.

Choice B is incorrect. This function represents a population in which each year the number of animals increased by 97%, not 3%, of the number of animals in the population the previous year.

Choice D is incorrect. This function represents a population in which each year the number of animals decreased, rather than increased, by 3% of the number of animals in the population the previous year.

Question 466 466 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

38x2=38(9)

What is the negative solution to the given equation?

Show Answer Correct Answer: -3

The correct answer is -3 . Dividing both sides of the given equation by 38 yields x 2 = 9 . Taking the square root of both sides of this equation yields the solutions x = 3 and x = -3 . Therefore, the negative solution to the given equation is -3 .

Question 467 467 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

y equals, a times x squared minus c

In the equation above, a and c are positive constants. How many times does the graph of the equation above intersect the graph of the equation y equals, a plus c in the xy-plane?

  1. Zero

  2. One

  3. Two

  4. More than two

Show Answer Correct Answer: C

Choice C is correct. It is given that the constants a and c are both positive; therefore, the graph of the given quadratic equation is a parabola that opens up with a vertex on the y-axis at a point below the x-axis. The graph of the second equation provided is a horizontal line that lies above the x-axis. A horizontal line above the x-axis will intersect a parabola that opens up and has a vertex below the x-axis in exactly two points.

Choices A, B, and D are incorrect and are the result of not understanding the relationships of the graphs of the two equations given. Choice A is incorrect because the two graphs intersect. Choice B is incorrect because in order for there to be only one intersection point, the horizontal line would have to intersect the parabola at the vertex, but the vertex is below the x-axis and the line is above the x-axis. Choice D is incorrect because a line cannot intersect a parabola in more than two points.

Question 468 468 of 479 selected Nonlinear Functions H

Let the function p be defined as p of x equals, the fraction with numerator, open parenthesis, x minus c, close parenthesis, squared, plus, 160, and denominator, 2 c , end fraction, where c is a constant. If p of c equals, 10, what is the value of p of 12 ?

  1. 10.00

  2. 10.25

  3. 10.75

  4. 11.00

Show Answer Correct Answer: D

Choice D is correct. The value of p(12) depends on the value of the constant c, so the value of c must first be determined. It is given that p(c) = 10. Based on the definition of p, it follows that:

p of c equals, the fraction with numerator open parenthesis, c minus c, close parenthesis, squared, plus 160, and denominator 2 c, end fraction, which equals 10

the fraction 160 over 2 c, end fraction, equals 10

2 c equals 16

c equals 8

This means that p of x equals, the fraction with numerator open parenthesis, x minus 8, close parenthesis, squared, plus 160, and denominator 16 for all values of x. Therefore:

p of 12 equals, the fraction with numerator open parenthesis, 12 minus 8, close parenthesis, squared, plus 160, and denominator 16

which equals the fraction with numerator 16 plus 160, and denominator 16

which equals 11

Choice A is incorrect. It is the value of p(8), not p(12). Choices B and C are incorrect. If one of these values were correct, then x = 12 and the selected value of p(12) could be substituted into the equation to solve for c. However, the values of c that result from choices B and C each result in p(c) < 10.

Question 469 469 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables M

x=8a(b+9)

The given equation relates the positive numbers a , b , and x . Which equation correctly expresses a in terms of b and x ?

  1. a=x8-(b+9)

  2. a=x8(b+9)

  3. a=8(b+9)x

  4. a=8x(b+9)

Show Answer Correct Answer: B

Choice B is correct. To express a in terms of b and x , the given equation can be rewritten such that a is isolated on one side of the equation. Since it’s given that b is a positive number, b + 9 is not equal to zero. Therefore, dividing both sides of the given equation by 8(b+9) yields the equivalent equation x8(b+9)=a, or a=x8(b+9).

Choice A is incorrect. This equation is equivalent to x=8(a+(b+9)).

Choice C is incorrect. This equation is equivalent to x=8(b+9)a.

Choice D is incorrect. This equation is equivalent to x=a8(b+9).

Question 470 470 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables E

the square root of x plus 4, end root, equals 11

What value of x satisfies the equation above?

Show Answer

The correct answer is 117. Squaring both sides of the given equation gives x plus 4, equals 11 squared, or x plus 4, equals 121. Subtracting 4 from both sides of this equation gives x equals 117.

Question 471 471 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

|-5x+13|=73

What is the sum of the solutions to the given equation?

  1. - 146 5

  2. -12

  3. 0

  4. 26 5

Show Answer Correct Answer: D

Choice D is correct. By the definition of absolute value, if |-5x+13|=73, then -5x+13=73 or -5x+13=-73. Subtracting 13 from both sides of the equation -5x+13=73 yields - 5 x = 60 . Dividing both sides of this equation by - 5 yields x = - 12 . Subtracting 13 from both sides of the equation -5x+13=-73 yields - 5 x = - 86 . Dividing both sides of this equation by - 5 yields x = 86 5 . Therefore, the solutions to the given equation are - 12 and 86 5 , and it follows that the sum of the solutions to the given equation is -12+865, or 26 5 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is a solution, not the sum of the solutions, to the given equation.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 472 472 of 479 selected Nonlinear Functions H

The function g is defined by g(x)=x(x-2)(x+6)2. The value of g(7-w) is 0 , where w is a constant. What is the sum of all possible values of w ?

Show Answer Correct Answer: 25

The correct answer is 25 . The value of g(7-w) is the value of g(x) when x=7-w, where w is a constant. Substituting 7-w for x in the given equation yields g(7-w)=(7-w)(7-w-2)(7-w+6)2, which is equivalent to g(7-w)=(7-w)(5-w)(13-w)2. It’s given that the value of g(7-w) is 0 . Substituting 0 for g(7-w) in the equation g(7-w)=(7-w)(5-w)(13-w)2 yields 0=(7-w)(5-w)(13-w)2. Since the product of the three factors on the right-hand side of this equation is equal to 0 , at least one of these three factors must be equal to 0 . Therefore, the possible values of w can be found by setting each factor equal to 0 . Setting the first factor equal to 0 yields 7-w=0. Adding w to both sides of this equation yields 7=w. Therefore, 7 is one possible value of w . Setting the second factor equal to 0 yields 5-w=0. Adding w to both sides of this equation yields 5=w. Therefore, 5 is a second possible value of w . Setting the third factor equal to 0 yields (13-w)2=0. Taking the square root of both sides of this equation yields 13-w=0. Adding w to both sides of this equation yields 13=w. Therefore, 13 is a third possible value of w . Adding the three possible values of w yields 7+5+13, or 25 . Therefore, the sum of all possible values of w is 25 .

Question 473 473 of 479 selected Nonlinear Functions E

f of x equals, open parenthesis, x plus 0 point 2 5 x, close parenthesis, times, open parenthesis, 50 minus x, close parenthesis

The function f is defined above. What is the value of f of 20 ?

  1. 250

  2. 500

  3. 750

  4. 2,000

Show Answer Correct Answer: C

Choice C is correct. Adding the like terms x and 0 point 2 5 x yields the equation f of x equals, 1 point 2 5 x times, open parenthesis, 50 minus x, close parenthesis. Substituting 20 for x yields f of 20 equals, open parenthesis, 1 point 2 5 times 20, close parenthesis, times, open parenthesis, 50 minus 20, close parenthesis. The product 1 point 2 5 times 20 is equal to 25, and the difference 50 minus 20 is equal to 30. Substituting these values in the given equation gives f of 20 equals, 25 times 30, and multiplying 25 by 30 results in f of 20 equals 750.

Choices A, B, and D are incorrect and may result from conceptual or computational errors when finding the value of f of 20.

 

Question 474 474 of 479 selected Equivalent Expressions E

Which expression is equivalent to x 2 + 3 x - 40 ?

  1. ( x - 4 ) ( x + 10 )

  2. ( x - 5 ) ( x + 8 )

  3. ( x - 8 ) ( x + 5 )

  4. ( x - 10 ) ( x + 4 )

Show Answer Correct Answer: B

Choice B is correct. The given expression may be rewritten as x2+8x-5x-40. Since the first two terms of this expression have a common factor of x and the last two terms of this expression have a common factor of -5 , this expression may be rewritten as x(x)+x(8)-5(x)-5(8), or x(x+8)-5(x+8). Since each term of this expression has a common factor of (x+8), it may be rewritten as (x-5)(x+8).

Alternate approach: An expression of the form x2+bx+c, where b and c are constants, can be factored if there are two values that add to give b and multiply to give c . In the given expression, b=3 and c=-40. The values of -5 and 8 add to give 3 and multiply to give -40 , so the expression can be factored as (x-5)(x+8).

Choice A is incorrect. This expression is equivalent to x2+6x-40, not x2+3x-40.

Choice C is incorrect. This expression is equivalent to x2-3x-40, not x2+3x-40.

Choice D is incorrect. This expression is equivalent to x2-6x-40, not x2+3x-40.

Question 475 475 of 479 selected Nonlinear Functions M

There were no jackrabbits in Australia before 1788 when 24 jackrabbits were introduced. By 1920 the population of jackrabbits had reached 10 billion. If the population had grown exponentially, this would correspond to a 16.2% increase, on average, in the population each year. Which of the following functions best models the population p of t of jackrabbits t years after 1788?

  1. p of t equals, 1 point 1 6 2 times, 24 raised to the t power

  2. p of t equals, 24 times, 2 raised to the 1 point 1 6 2, t power

  3. p of t equals, 24 times, 1 point 1 6 2 raised to the t power

  4. p of t equals, open parenthesis, 24 times 1 point 1 6 2, close parenthesis, raised to the t power

Show Answer Correct Answer: C

Choice C is correct. This exponential growth model can be written in the form p of t equals, A, times, open parenthesis, 1 plus r, close parenthesis, raised to the t power, where p of t is the population t years after 1788, A is the initial population, and r is the yearly growth rate, expressed as a decimal. Since there were 24 jackrabbits in Australia in 1788, A, equals 24. Since the number of jackrabbits increased by an average of 16.2% each year, r equals 0 point 1 6 2. Therefore, the equation that best models this situation is p of t equals, 24 times, 1 point 1 6 2 raised to the t power.

Choices A, B, and D are incorrect and may result from misinterpreting the form of an exponential growth model.

 

Question 476 476 of 479 selected Equivalent Expressions E

Which of the following expressions is equivalent to the sum of open parenthesis, r cubed, plus 5 r squared, plus 7, close parenthesis and open parenthesis, r squared, plus 8 r, plus 12, close parenthesis ?

  1. r to the fifth power, plus 13 r cubed, plus 19

  2. 2 r cubed, plus 13 r squared, plus 19

  3. r cubed, plus 5 r squared, plus 7 r, plus 12

  4. r cubed, plus 6 r squared, plus 8 r, plus 19

Show Answer Correct Answer: D

Choice D is correct. Grouping like terms, the given expressions can be rewritten as r cubed plus, open parenthesis, 5, r squared plus r squared, close parenthesis, plus 8 r, plus, open parenthesis, 7 plus 12, close parenthesis. This can be rewritten as r cubed, plus 6, r squared, plus 8 r, plus 19.

Choice A is incorrect and may result from adding the two sets of unlike terms, r cubed and r squared as well as 5, r squared and 8 r, and then adding the respective exponents. Choice B is incorrect and may result from adding the unlike terms r cubed and r squared as if they were r cubed and r cubed and adding the unlike terms 5, r squared and 8 r as if they were 5, r squared and 8, r squared. Choice C is incorrect and may result from errors when combining like terms.

 

Question 477 477 of 479 selected Equivalent Expressions E

Which expression is equivalent to 12 x + 27 ?

  1. 12(9x+1)

  2. 27(12x+1)

  3. 3(4x+9)

  4. 3(9x+24)

Show Answer Correct Answer: C

Choice C is correct. Each term in the given expression, 12 x + 27 , has a common factor of 3 . Therefore, the expression can be rewritten as 3(4x)+3(9), or 3(4x+9). Thus, the expression 3(4x+9) is equivalent to the given expression.

Choice A is incorrect. This expression is equivalent to 108 x + 12 , not 12 x + 27 .

Choice B is incorrect. This expression is equivalent to 324 x + 27 , not 12 x + 27 .

Choice D is incorrect. This expression is equivalent to 27 x + 72 , not 12 x + 27 .

Question 478 478 of 479 selected Nonlinear Functions H

A right rectangular prism has a height of 9 inches. The length of the prism's base is x inches, which is 7 inches more than the width of the prism's base. Which function V gives the volume of the prism, in cubic inches, in terms of the length of the prism's base?

  1. V(x)=x(x+9)(x+7)

  2. V(x)=x(x+9)(x-7)

  3. V(x)=9x(x+7)

  4. V(x)=9x(x-7)

Show Answer Correct Answer: D

Choice D is correct. The volume of a right rectangular prism can be represented by a function V that gives the volume of the prism, in cubic inches, in terms of the length of the prism's base. The volume of a right rectangular prism is equal to the area of its base times its height. It's given that the length of the prism's base is x inches, which is 7 inches more than the width of the prism's base. This means that the width of the prism's base is x - 7 inches. It follows that the area of the prism's base, in square inches, is x(x-7) and the volume, in cubic inches, of the prism is x(x-7)(9). Thus, the function V that gives the volume of this right rectangular prism, in cubic inches, in terms of the length of the prism's base, x , is V(x)=9x(x-7).

Choice A is incorrect. This function would give the volume of the prism if the height were 9 inches more than the length of its base and the width of the base were 7 inches more than its length.

Choice B is incorrect. This function would give the volume of the prism if the height were 9 inches more than the length of its base.

Choice C is incorrect. This function would give the volume of the prism if the width of the base were 7 inches more than its length, rather than the length of the base being 7 inches more than its width.

Question 479 479 of 479 selected Nonlinear Equations In 1 Variable And Systems Of Equations In 2 Variables H

x 2 + y + 7 = 7

20x+100-y=0

The solution to the given system of equations is (x,y). What is the value of x ?

Show Answer Correct Answer: -10

The correct answer is -10 . Adding y to both sides of the second equation in the given system yields 20 x + 100 = y . Substituting 20 x + 100 for y in the first equation in the given system yields x2+20x+100+7=7. Subtracting 7 from both sides of this equation yields x 2 + 20 x + 100 = 0 . Factoring the left-hand side of this equation yields (x+10)(x+10)=0, or (x+10)2=0. Taking the square root of both sides of this equation yields x + 10 = 0 . Subtracting 10 from both sides of this equation yields x = -10 . Therefore, the value of x is -10 .